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Investigation of electric quadrupole strength in ¹³N using the ¹²C(p,Ύ₀)¹³N reaction Helmer, Richard Lloyd 1977

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INVESTIGATION OF ELECTRIC „QUADRUPOLE STRENGTH IN i*N USING THE i 2 C ( p # Y 0 ) 13N REACTION by RICHARD LLOYD HELM EH B.A.Sc, University of B r i t i s h Columbia, 1966 H, A. Sc., Oniversity of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF r DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES {Department of Physics) We accept t h i s t h e s i s as conforming to the reguired standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 ©Richard Lloyd Helmer, 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the 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 r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f 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 g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date W f t 1111 ABSTRACT The E2 cross section for the l 2 C { p , Y Q ) i 3 H reaction has been measured from 10 MeV to 17 Me? in the laboratory system by bombarding an enriched carbon-12 target with beams of polarized protons. A 10 i n . , * x 10 i n , Nal(Tl) detector with a p l a s t i c anti-coincidence s h i e l d was used to detect the gamma rays. The t o t a l E2 capture cross sections were of the order of 0.2 ybarns and no resonance e f f e c t s were observed. The amount of the E2 energy-weighted sum rule depleted i n t h i s energy range i s (10.3 ± 4.0)%. Calculations based on a d i r e c t semi-direct capture model provide a good description of the experimental r e s u l t s by including only d i r e c t E2 capture and di r e c t plus c o l l e c t i v e E1 capture. i i i TABLE OF CONTENTS ABSTRACT ....................... i i TABLE OF CONTENTS i i i LIST OF TABLES V LIST OF FIGURES v i ACKNOWLEDGEMENTS . v i i i CHAPTER I Introduction , 1 1.1 General Introduction 1 1.2 Review of Previous Work 9 1.3 Present Work ..... 10 CHAPTER II Experimental Apparatus and Procedure ......... 13 2.1 General Experimental Arrangement 13 2.2 Gamma Ray Spectrometer ...17 2.3 Gamma Spectrometer Electronics ...............18 2.4 P a r t i c l e Detection 24 2.5 Targets 27 2.6 Current Integration 29 2.7 Polarization Measurements 30 2.8 Data Accumulation ............................ 30 CHAPTER III Data Analysis and Results .................... 33 3.1 Gamma Ray Spectra Analysis ................... 33 3.2 P a r t i c l e Spectra Analysis .......37 3.3 Results from the Pa r t i c l e Analysis 38 .3.4 Results from the Gamma Ray Analysis 40 3.5 Beam Polarization Measurements ............... 46 3.6 Angular D i s t r i b u t i o n Functions ................ 48 3.7 Results of the Angular Distribution F i t s 54 3.8 Extraction of the T-matrix Elements .......... 58 3.9 Determination of the Cross Sections .......... 73 CHAPTER IV The DSD Model, Sum Rules and Comparison with Other Experiments 77 4.1 The DSD Model 78 4.2 The Op t i c a l Model and Calculation of the Have Functions .86 4.3 Calculation of the Direct Semi-Direct capture 96 4.4 Sum Rules ...........................102 4.5 Comparison with Other Work ......103 CHAPTER V Summary and Conclusions .....109 BIBLIOGRAPHY .... ...113 APPENDIX A The Energy Weighted Sum Rule ...121 APPENDIX B T-matrix Elements ., ...................124 APPENDIX C Polarized Proton Beam Asymmetries ..128 V LIST OF TABLES Table III-1 Summary of Beam Pol a r i z a t i o n Measurements ..,.,47 Table III-2 Relation between the Angular Distribution C o e f f i c i e n t s and the Reaction Matrix Elements 53 Table III-3 Solid Angle Correction Factors ............... 54 Table III-4 l 2 C (p, Y Q ) 13N Angular Distribution C o e f f i c i e n t s ................................. 57 Table III-5 T-matrix Element F i t s to I 2 C ( p , ^ Q ) 1 3 N Angular Distributions - Solution I ...................61 Table III-6 T-matrix Element F i t s to 1 2 C ( p , Y Q ) 1 3 H Angular Distributions - Solution II .,..,62 Table III-7 E1 Amplitude Ratio s/d and Parameters Related to E2 Capture for Solution I ................. 65 Table III-8 second Solutions to the T-matrix Element F i t s 69 Table IV-1 Optical Model Parameters ..................... 90 Table IV-2 GDR Parameters Dsed to Reproduce the Total Cross Section Using the WSS Potential ........100 Table B-1 Relation between the Angular Distribution C o e f f i c i e n t s and the Reduced T-matrix Elements 125 Table B-2 Relations between the Angular Distribution C o e f f i c i e n t s and the Reaction Amplitudes .,,..126 v i LIST OF FIGURES Figure II-1 Schematic view of the beam l i n e ............. 14 Figure II-2 Block diagram of the gamma spectrometer el e c t r o n i c s ................................. 20 Figure II-3 i a C { p , y Q ) 1 3 N gamma ray spectrum at E+ = 11.2 HeV 23 p Figure II-4 Block diagram of the p a r t i c l e detector electronics .................................26 Figure II-5 P a r t i c l e spectrum at E£ = 11.2 MeV .......... 28 Figure III-1 Reject to accept r a t i o s f o r E^ = 10 MeV ..... 36 Figure III-2 Polarized proton beam asymmetries at E-* = 15 MeV and 16 MeV 39 P Figure III-3 The complete angular d i s t r i b u t i o n measurements at E+ = 10 MeV 43 Figure III-4 D i s t r i b u t i o n of x 2 for the l 2 C ( p r ^ 0 ) l 3 N yields and asymmetries ...................... 45 Figure III-5 1 2 C (p , Y 0 ) i 3 j * normalized d i f f e r e n t i a l cross sections 55 Figure III-6 1 2C-tp, l 3N angular d i s t r i b u t i o n s f o r the asymmetries ............56 Figure III-7 1 2 C { p , Y 0 ) i 3 f j normalized angular d i s t r i b u t i o n c o e f f i c i e n t s 59 Figure III-8 E1 amplitudes and re l a t i v e phase 63 v i i Figure 111-9 The amplitude r a t i o and phases related to E2 capture 66 Figure III- 1 0 Projection of the multidimensional x 2 ~ s u r f a c e onto the E2 strength axis . 6 8 Figure I I I - 1 1 The normalized a and angular d i s t r i b u t i o n c o e f f i c i e n t s 71 Figure 'III-12 The E 2 cross sections . 7 4 Figure IV -1 Schematic representation of dire c t and semi-direc t processes 8 0 Figure IV-2 1 2 C ( P , p Q ) 1 2 C d i f f e r e n t i a l cross section and analyzing power at E+ = 1 6 . 9 6 4 MeV . . . . . . . . . . 9 3 Figure IV-3 Comparison of the o p t i c a l model analyses with some experimental p a r t i a l reaction cross sections .................................... 94 Figure IV-4 The r e a l parts of the r a d i a l wave functions calculated with the NEW potential 97 Figure IV - 5 F i t s to the 1 2 C ( p , Y 0 ) i 3 N t o t a l cross section 9 8 v i i i ACKNOWLEDGEMENTS I owe sincere appreciation to my supervisor. Dr. Mike Hasinoff, for the judicious prodding which has brought me to the end of my graduate student days. Refocussing my attention was made necessary by my many drinking friends at the C e c i l Hotel, s a i l i n g friends at the K i t s i l a n o Yacht Club, hockey friends at the Winter Sports Centre and school f r i e n d s at the Universities of B r i t i s h Columbia and Washington who a l l distracted me from my studies. I wish to thank the s t a f f at the Nuclear Physics Laboratory of the University of Washington for t h e i r very f r i e n d l y and helpful assistance during the course of the experimental measurements. I wish es p e c i a l l y to thank John B u s s o l e t t i , Katsu Ebisawa and Dr. Kurt Snover for many helpful and stimulationg discussions about t h i s work, and Dr. P h i l Dickey for putting up with me during my stays i n Seattle. I would have thanked Lee at t h i s point for a l l the love and encouragement she has given me and for the tremendous amount of work she did in helping to prepare t h i s t h e s i s , but she preferred cash instead! 1 Chapter I INTRO DDCTIQN 1.1 General Introduction Photonuclear reactions are a r e l a t i v e l y simple way to obtain some of the d e t a i l s of the structure of the nucleus. The s i m p l i c i t y arises because the electromagnetic operator which mediates the interaction i s r e l a t i v e l y weak, so that perturbation theory can be used with some confidence to describe the e f f e c t s of the in t e r a c t i o n . In addition, the electromagnetic operator i s well understood, so that the information acguired i s d i r e c t , in the sense that no a p r i o r i knowledge of the les s well known nuclear force need be assumed. One of the most f r u i t f u l of the types of photonuclear reactions to be studied has been the excitation and decay of the Giant Dipole Resonance (GDR). This resonance, which was f i r s t shown to exis t i n the late f o r t i e s (BA 47), i s characterized by three basic properties (FU 73) . F i r s t , i t exists i n a l l nuclei at an excitation energy which varies from approximately 80A~*A MeV for the heavy nuclei to approximately .5OA-0 HeV for the l i g h t e r n u c l e i . Second, i t has great strength in that i t exhausts s l i g h t l y in excess of the c l a s s i c a l dipole sum r u l e . This sum rule was f i r s t derived for 2 nuclei by Levinger and Bethe {LE 50), and gives a conservation law for the integrated absorption cross section. Neglecting exchange and ve l o c i t y dependent forces, the sum rule i s given F i n a l l y , the dipole strength i s concentrated in a r e l a t i v e l y narrow energy region, the width varying from about 3 MeV for closed s h e l l n u c l e i to about 9 MeV for deformed n u c l e i . These l a s t two properties combine to give the GDR i t s resonance shape. The GDR i s viewed i n the c o l l e c t i v e model as a bulk o s c i l l a t i o n of a l l the protons i n the nucleus moving against a l l the neutrons i n the nucleus {GO 48 and ST 50). In s h e l l model language, the resonance i s formed by the action of the incoming gamma ra i s i n g a nucleon to the next higher major s h e l l {WI 56). This would imply that the energy of the GDH should be 1 -hoi {or about 41A - 1/ 3 MeV) which i s too low. Brown and B o s t e r l i (BR 59a) pointed out that the hole that i s l e f t behind i n t h i s process must be strongly correlated i n angle with the excited nucleon because the two are coupled to an angular momentum and parity of 1 _. These authors then showed that the GDR i s constructed from a coherent superposition of these parti c l e - h o l e states. Both the c o l l e c t i v e model and s h e l l model descriptions of the GDR have the i r l i m i t s of a p p l i c a b i l i t y (SP 69), and both have had many extensions and refinements to improve the agreement between theory and experiment (see, for example, DA 65 and SP 69). by MeV-barns 1-1 3 In the past few years, evidence of a resonance other than the GDR has come to l i g h t . This was f i r s t seen in i n e l a s t i c electron scattering data (PI 71) and i n reexamination (LE 72) of e a r l i e r i n e l a s t i c proton scattering data (TY 58), both of which showed a new resonance located 2 to 3 HeV below the GDR. These early studies, as well as more recent ones (BE 76a), strongly suggest that this new resonance i s e l e c t r i c guadrupole (E2) i n character, and hence i t has come to be c a l l e d the Giant Quadrupole Resonance (GQR). The GQR had been expected for some time before i t s discovery, since the e f f e c t i v e charges needed to explain e l e c t r i c quadrupole t r a n s i t i o n rates and moments depended e x p l i c i t l y on some of the E2 strength to be l y i n g at high excitation energies (BO 69a). The s h e l l model depicts the GQR as a superposition of particle-hole states in which the p a r t i c l e s have been excited through two major s h e l l s . Two modes of coherent motion are possible, one i n which the neutrons and protons move i n phase (isoscalar) and one in which they move out of phase fisovector). Because the interaction between the nucleons i s at t r a c t i v e in the is o s c a l a r mode, the resonance energy i s pulled down from the expected value of 2 nto. . Bohr and Mottelson have shown, on quite general grounds, that the expected energy of the isoscalar GQR i s 5 8A~V3 HeV (BO 69b) , and th i s i s approximately the observed resonance energy. There should also be an isovector part to the GQR, which, because of the repulsive nature of the inte r a c t i o n between nucleons i n th i s mode, i s expected to l i e at higher excitation energies. F i n a l l y , e x c i t a t i o n of particle-hole states within a major s h e l l contribute to c o l l e c t i v e , i s oscalar E2 strength. These states correspond to the low-lying 2 + states of even-even n u c l e i . In nuclei heavier than *°Ca, the GQR seems to be lo c a l i z e d enough to appear as a resonance, with about 80 or 90 per cent of the Gell-Mann-Telegdi (GE 53) energy-weighted sum rule (EWSR) being depleted. This sum rule i s a conservation law for iso s c a l a r E2 tr a n s i t i o n s that i s e s s e n t i a l l y model independent. The EWSR i s given by where <r z> i s the mean sguared displacement from the centre-of-mass of a nucleon in the ground state of the nucleus, and E i s the energy. The expression 1-2 i s developed i n appendix A. The peak energy of the GQR l i e s at approximately 63A-1/3 MeV for nuclei in the range 40<A<120, possibly a l i t t l e higher than t h i s for nuclei with higher mass numbers and a l i t t l e lower for nuclei with lower mass numbers (BE 76a). The width of the resonance i s smallest for closed s h e l l nuclei, and decreases from 7 HeV for the l i g h t e r nuclei to 3 MeV for the heavier. Only about 30% of the EWSR i s exhausted in the giant resonance region f o r nuclei with A<40. This i s partly due to the fact that more of the strength apparently resides i n the low^lying bound states of l i g h t nuclei than heavy nu c l e i . A study of i n e l a s t i c electron scattering on *°Ca showed that the E2 strength was rather uniformly spread between 10 and 1-2 5 20 MeV <TO 73). Spreading of the guadrupole strength has also been seen i n i n e l a s t i c electron scattering studies on **0 (HO 74), where 43% of the E8SR l i m i t was found below E x = 20 MeV. ; Further examples of t h i s e f f e c t have been seen i n many radia t i v e alpha capture reactions. For example, a capture reactions on * 2 C (SN 74), 2 * i 2 6 f t g and 2®Si (ME 68), 3*Ar (Hfl 73) , and several other nuclei (KD* 74) , show that a considerable f r a c t i o n of the sum rule i s exhausted below 63A-1/3 MeV, and that the strength i s spread from this energy down to the f i r s t excited 2 +-state (in even-even nuclei). However, i n e l a s t i c alpha s c a t t e r i n g studies on *°Ca (RU 74) and several other l i g h t nuclei (KN 76) do show evidence of a GQR which exhausts about 30$ of the EWSR. A si m i l a r i n e l a s t i c alpha scattering measurement on 1 2 C (KN 76) showed no evidence of a GQR although a small amount (6 ± 2%) of the EwSR was seen near E x = 27 MeV. This r e s u l t i s consistent with a recent continuum s h e l l model c a l c u l a t i o n for (BI 75) which predicted a GQR at about t h i s energy, but which was expected to be guite broad (r>>5 MeV) because of coupling to low-lying c o l l e c t i v e states (KN 76). Thus i t now appears that a resonance structure persists perhaps down to A = 16 MeV, but for n u c l e i as l i g h t as l 2C, the resonance has either disappeared or has been washed out because of broadening. Good resolution (~150 keV) alpha p a r t i c l e scattering from i^o showed a number of peaks i n the expected GQR region that were assigned to L=2 transfer (HA 7 6) , the sum of which exhausted about 401 of the E8SR. Similar peaks have been observed with i n e l a s t i c proton sc a t t e r i n g i n the giant 6 resonance region of l 2C (GE 75), although these findings were not confirmed i n the i n e l a s t i c alpha scattering measurements of Knopfle et a l . (K» 76).. Thus, i n addition to the spreading of the guadrupole strength in l i g h t n uclei, there i s some evidence for i t to be fragmented. This makes experimental observation more d i f f i c u l t , and may help to explain why so much less of the EWSR i s observed in l i g h t nuclei than in heavy nuclei., among the heavier n u c l e i , only zoapb seems to exhibit any s i m i l a r fine structure (MO 76) . In general, theories of the giant resonances predict l i t t l e more than t o t a l cross sections and strength d i s t r i b u t i o n s . For example, calculations based on the bound partic l e - h o l e excitations of Brown and B o s t e r l i {BR 59a) can do no more than describe the gross shape of the GDR. When more complicated configurations are introduced to describe the intermediate structure, the calculations become much more d i f f i c u l t and the physical e f f e c t s tend to become obscured. I t i s also necessary to include the e f f e c t s of the continuum to describe the s i t u a t i o n properly. Such calculations have been done, but they have not met with t o t a l success. For example, Wang and Shakin (WA 72) included both the above ef f e c t s to describe the intermediate structure seen i n the photodisintegration of * 60. I t was found that f a i r l y large phenomenological E2 amplitudes were then reguired to f i t the neutron polarization data of Cole et a l . (CO 69). However, measurements of polarized proton capture in 1 SN by Hanna et a l . (HA 74a) found that the E2 amplitudes were much less than predicted. Thus the t h e o r e t i c a l description of the giant 7 resonances i s not yet in a s a t i s f a c t o r y state. The present work i s a study of guadrupole absorption i n l 3N via the inverse reaction, radiative polarized proton capture on 1 2C. The inverse t 3 N ( Y , p o ) 1 2 C reaction i s related to the one studied by the p r i n c i p l e of detailed balance. An expression r e l a t i n g the two reactions i s given i n Appendix A. The reason for using polarized rather than unpolarized protons i s that physically independent information i s obtained on the interference between the various p a r t i a l waves taking part i n the reaction (GL 73)., Thus there are more constraints available to help extract the parameters of interest. Studies with capture reactions suffer from several disadvantages. F i r s t , guadrupole radiation i s 10 to 10 0 times l e s s intense than dipole radiation, and therefore i t i s d i f f i c u l t to observe d i r e c t l y . Second, information concerning giant resonances b u i l t on the ground state of the residual nucleus i s a l l that can be obtained, although i n i s o l a t e d cases i t should also be possible to obtain information about the giant resonances b u i l t on low-lying excited states of the residual nucleus. Third, i t may happen that the GQR of the nucleus under consideration p a r t i c l e decays to high-lying excited states i n the residual nuclei, so very l i t t l e strength w i l l appear i n the ground state channels. However, the E2 strength that i s seen can often be extracted with great confidence, since the interpretation of angular d i s t r i b u t i o n and p o l a r i z a t i o n measurements has been well developed. In addition, the backgrounds underlying the peaks of interest in the y-ray spectra are much les s severe and 8 much better understood than the continuum underlying the peaks in i n e l a s t i c scattering spectra. F i n a l l y , for the p a r t i c u l a r case being considered, 1 3N beta decays to i 3 C with a h a l f l i f e of about 10 minutes (AJ 70) and therefore the giant resonances i n 1 3N can be studied d i r e c t l y only via radiative capture reactions. Most previous (p,'Y) {the adoptions of the Madison Convention {BA 70) are used throughout t h i s thesis) studies have concentrated on learning more detai l s about the GDR. The f i r s t such measurement was made by Glavish et a l . {GL 72) who studied the 1 J B ( p , Y Q ) t z c reaction. The GDR i n other nuclei, for example *He (GL 73), «ozr {HA 73a), zo^e (GL 73) and * a s i (GL 73) have also been investigated using t h i s technique. The above mentioned studies were a l l carried out by the Stanford group, who also made the f i r s t extensive study of E2 strength with the (p, Y 0) reaction. The r e s u l t s of t h e i r work on the l sN (p,"f0) 1 6 0 reaction have already been discussed b r i e f l y . They found evidence for a GQR in the (Y,p o) channel which exhausted 30% of the EWSR between E =20 MeV and 26.5 x MeV. Recently, t h i s reaction and the l 4 C ( p , Y 0 ) l s N reaction have been studied at the Oniversity of Washington (AD 77, BU 76a). This work w i l l be described more f u l l y i n l a t e r chapters where comparisons to the present experiment w i l l be made. . 9 1.2 Review of Previous Work Previous experimental work on 1 2 C ( p , Y 0 ) i 3 u in the giant resonance region has centred on extracting the d e t a i l s of the GDH. The f i r s t measurements of the ground state gamma radiation in the giant resonance region were obtained over a very limited energy range by warburton and Funsten (wA 62). Fisher et a l . (FI 63) l a t e r investigated the region from E p = 11 MeV to 39 MeV. Both measurements were hampered by poor beam energy resolution and by poor detector energy resolution; nevertheless, the gross features of the GDR were elucidated and a f i r s t e f f o r t (ME 65a) at describing the mechanisms involved i n i t s ex c i t a t i o n was made by comparison to the s h e l l model cal c u l a t i o n s of Barker (BA 61) and Easlea (EA 62). In order to learn more about the d e t a i l s of the low energy "pygmy" resonance seen at E„ ~14 MeV i n the e a r l i e r measurements, Measday et a l . (ME 73a) measured the 90* y i e l d curve from E p = 8.6 MeV to 16.0 MeV with much improved resolution. Thus they were able to observe some intere s t i n g features in t h i s region, including two dramatic interference dips at 10.6 MeV and 13.1 MeV. The 90® y i e l d curve was l a t e r extended to E p =24.4 MeV by Berghofer e t - a l * ,(BE 76b) who found that the main strength of the GDR seen i n the {p, Y 0 ) channel was centred at E x = 20.8 MeV with a width of 4 MeV. In addition, the y i e l d curve i n the experimentally d i f f i c u l t region from E p = 3 MeV to 9 MeV has been measured by Johnson (JO 74) . Angular d i s t r i b u t i o n s were measured at several incident 10 proton energies between 10 and 2H MeV by Berghofer et a l . A Legendre polynomial expansion of the angular d i s t r i b u t i o n s reguired the presence of odd terms to f i t the data, and i t w i l l be shown i n Chapter III that the odd terms aris e from E2 radiation. Thus there was evidence in these measurements that E2 radiation was present and was i n t e r f e r i n g with the dominant E1 ra d i a t i o n , but no guantitative estimates could be made. Evidence of E2 radiation i n the giant resonance region of the stable mirror nucleus 4 3C was seen i n the i n e l a s t i c electron scattering data of Shin et a l . (SH 71), but again no guantitative estimates were made. This l a t t e r measurement w i l l be discussed more f u l l y i n Chapter IV. Further evidence of E2 strength i n 1 3 C was seen by Arthur, Drake and Halpern (AH 75). These authors studied r a d i a t i v e neutron capture by 1 2 C at an excitation energy in 1 3 C of 18 MeV, and found non-zero odd Legendre polynomial c o e f f i c i e n t s i n the angular d i s t r i b u t i o n . It can be shown that radiative neutron capture i s more sensi t i v e than radiative proton capture to c o l l e c t i v e E2 strength (HA 73b), so t h i s measurement gives strong evidence f o r a possible GQB in 1 3 C (although i t was not clear whether t h i s strength was isoscalar or isovector i n character). 1.3 Present Work It can be measurement of the seen from the previous section that a E2 strength in the giant resonance region i s 11 a natural extension of the work already done. I t i s not usually possible to extract the E2 amplitudes unambiguously even from a polarized proton capture experiment. However, for the p a r t i c u l a r instance of polarized spin 1/2 p a r t i c l e s incident on a spin 0 (or spin 1/2) nucleus, as i s the case here, these amplitudes can, in p r i n c i p l e , be uniquely obtained (provided M1 r a d i a t i o n can be neglected). Therefore, a study of i 2C(p,Y 0) 13{j i s useful because i t may be one of those cases mentioned e a r l i e r where the E2 strength can be confidently extracted. Recently, however, considerable ambiguity has been found i n the interpretation of even these simple experiments (BU 76b). These ambiguities include finding double solutions to the E2 cross sections derived from the data. This d i f f i c u l t y , and others, w i l l be discussed more f u l l y i n l a t e r chapters where comparisons w i l l be made to the present r e s u l t s . The fact that i t might be possible to r e l i a b l y determine the guadrupole strength provides a second independent reason to study t h i s reaction. I t was mentioned e a r l i e r i n the introduction that d i f f i c u l t i e s are encountered when attempts are made to c a l c u l a t e t h e o r e t i c a l l y the properties of the giant resonances. In order to ameliorate the s i t u a t i o n , at least temporarily, i t i s necessary to resort to reaction models to help distinguish among the possible alternatives. Reaction models provide a connection between the parameters of the states concerned and the experimentally observed quantities. Such a reaction model i s being developed by Snover and Ebisawa (SN 75) to help understand the E2 strength seen i n ra d i a t i v e 12 capture reactions. This model i s based on the d i r e c t semi-direct capture (DSD) model f i r s t proposed by Brown to explain E1 cross sections near the GDR (BR 64). The reaction * 2 C ( p , Y 0 ) i 3 { j should provide a good test of t h i s reaction model. The model and comparisons to the data presented here w i l l be described in Chapter IV. The energy range covered in t h i s experiment i s from 10 MeV to 17 MeV incident proton energy. This range of energies approximately covers the "pygmy" resonance observed i n the 90° y i e l d curve and i n the t o t a l cross section (ME 73a, BE 76b). This resonance also appears i n the 90° y i e l d curve of the 1 3C(Y,n )««C reaction (JO 77), and i n the 1 3C{Y,n) 1 2C data of Koch and Thies (KO 76). Below 10 MeV, the dipole strength begins to f a l l r a p i d l y . The guadrupole strength presumably f a l l s off even more rapidly, except f o r the possible presence of i s o l a t e d narrow states., The upper l i m i t of 17 MeV was dictated by the maximum beam energy available. An account of the apparatus and measurement techniques used i n t h i s experiment i s given in Chapter I I . The methods of data analyses and the r e s u l t s are presented i n Chapter I I I . Comparison of the re s u l t s for the 1 2C(p,Y 0)* 3N reaction to other experiments and to the EWSR are given in Chapter IV, i n addition to a description of the attempts to f i t the data with the DSD model. Chapter V contains a summary of the r e s u l t s and the conclusions. 13 Chapter II EXPERIMENTAL APPARATUS AND PROCEDURE The main objective of t h i s experiment was to measure the E2 cross section as a function of energy for the reaction * 2C (p, YQ) 1 3N. In addition, i t was desirable to measure, as accurately as possible, polarized and unpolarized angular d i s t r i b u t i o n c o e f f i c i e n t s and various other parameters related to the p a r t i a l waves taking part i n the reaction. A l l the quantities extracted from the data could then be compared to the reaction model of Snover and Ebisawa. This chapter contains a description of the eguipment and procedures used to c o l l e c t the data., 2.1 General Experimental Arrangement A schematic diagram of the experimental set-up i s given i n Figure IT-1. The incident polarized proton beam was produced by the University of Washington Lamb-shift Polarized Ion Source (FA 71). The beam was then accelerated by the University of Washington FN tandem Van de Graaff accelerator, bent through a 90° analyzing magnet and directed down the appropriate beam l i n e (30°) by a switching magnet. The beam was magnetically focussed through a collimator and skimmer system onto the target. After passing through the target, the beam travelled 15 another 7 m further downstream u n t i l i t reached the beamstop located behind concrete and wax shielding. The long downstream beam tube served as the Faraday cup. The beam l i n e was o p t i c a l l y aligned by viewing through a telescope mounted at the downstream end and focussing on a cross-hair located at the e x i t of the switching magnet. The collimator and skimmer, which are located in a separate section of beam l i n e , were then accurately centred by s h i f t i n g t h i s section the necessary amount. F i n a l l y , a cross-wire was mounted in the centre of the target chamber and the chamber was moved u n t i l the cross-wire was centred in the beam l i n e . At the same time, a plumb bob was used to check that the cross-wire was v e r t i c a l l y above the centre axis of the gamma ray angular d i s t r i b u t i o n table. The spin orientation of the polarized proton beam was changed by reversing the guench and argon f i e l d s . This was controlled by means of a f l i p p e r described by Adelberger jgt aJL. (AD 73). The pol a r i z a t i o n could be f l i p p e d automatically from one to ten times a second, or i t could be fli p p e d manually when desired. The f l i p p e r also provided a l o g i c routing signal which was used to route other signals according to whether the proton spin was up or down. The amount of beam on target was limited by the counting rate in the Nal(Tl) detector. Currents from 30 namps to 60 namps were sa t i s f a c t o r y , depending on the beam energy. The beam s t r i k i n g the collimators was continuously monitored and was t y p i c a l l y 0.2 narap. I f the collimator current rose as high as 2 namps, the experiment was stopped and the beam refocussed. 16 This was necessary to prevent the appearance of troublesome backgrounds i n the spectrum. The target holder was a ladder on which three targets could be mounted. One of these was an aluminum blank with the same diameter hole as the actual target. The ladder could be rotated to any orientation about a v e r t i c a l axis. This was useful in that i t allowed the target holder and ladder frame to be turned out of the l i n e of sight of the gamma detector. The target holder was surrounded by a copper cylinder which was maintained at l i g u i d nitrogen temperature. This helped to reduce any buildup of beam l i n e contaminants on the target. A s l o t was cut out of the middle of the cylinder to allow the beam and scattered protons to pass through f r e e l y . A copper s t r i p 0 . 0 0 2 inches thick was soldered over the s l o t where the gamma rays passed to the detector. This produced v i r t u a l l y no attenuation of the gamma f l u x , but did improve the vacuum i n the immediate v i c i n i t y of the target. Undesirable backgrounds can arise from beam s t r i k i n g the aluminum target frame or the copper cold-trap. However, a collimator of diameter 3/16 inch and skimmer of diameter 1/4 inch seemed to be small enough to prevent t h i s happening. Checks were made at various times by passing the beam through the aluminum blank in the target holder. No e l a s t i c a l l y scattered protons were observed in the p a r t i c l e spectra under these conditions. 17 2.2 Gamma Hay Spectrometer The most important instrument used i n this experiment was the gamma spectrometer. I t consisted of a large central Nal(Tl) detector surrounded by a p l a s t i c anti-coincidence (AC) shi e l d . This spectrometer has been described i n d e t a i l elsewhere (HA 74b), and general considerations for the design of such spectrometers have been given by Paul (PA 74), so only the s a l i e n t features w i l l be described here. A view of the spectrometer i s shown in Figure II-1. The central c r y s t a l i s in the form of a cylinder 25.4 cm i n diameter by 25.4 cm long. I t i s viewed by seven EMI 9758B photon u l t i p l i e r s . The surrounding anti-coincidence s h i e l d , manufactured from the p l a s t i c s c i n t i l l a t o r NE 110, i s 10.8 cm thick. I t covers the sides and front face of the central c r y s t a l . The cylinder i s viewed by six phototubes and the front p l a s t i c by two phototubes (HCA 8055). The space between the two detectors i s f i l l e d with a 1 cm thick self-supporting mixture of lithium carbonate and wax (LI 75). This helps to reduce the background due to slow neutron capture in the central c r y s t a l . The sides of the entire assembly are surrounded by 4 inches of lead to reduce the cosmic ray flux reaching the central c r y s t a l . The front i s also shielded by 4 inches of lead. This reduces the low energy gamma background reaching the detector from the target. The front shielding has provision f or di f f e r e n t size collimators to be inserted. The 18 resolution of the spectrometer i s improved with smaller collimators, but since t h i s was not of paramount importance i n the present experiment, the inse r t i o n hole, with a diameter of 6 inches, was l e f t completely open. The gamma spectrometer was located s u f f i c i e n t l y f a r from the beam l i n e - about 16 inches from the centre of the target to the front face of the lead shielding - that i t could be swung through angles from 43° to 137°. It would have been desirable, of course, to have had a larger angular range i n order to reduce the errors i n the experiment. However, this would have meant having the detector further back with a consequent decrease in counting rate. The constancy of the distance from the detector to the centre of the chamber was checked by mounting a RaTh source i n the target holder and measuring the "angular d i s t r i b u t i o n " as the detector was swung through i t s range., The counts recorded were i s o t r o p i c to within 1/4%. This test also ensured that absorption through the chamber walls was uniform. 2.3 Gamma Spectrometer Electronics A f a i r l y complex, but now b a s i c a l l y standard, system for processing the sign a l s from the gamma spectrometer was u t i l i z e d . , The fundamental idea behind the e l e c t r o n i c system i s to veto events which are affected by pile-up, and to reject cosmic ray events and events for which some of the energy escapes from the central c r y s t a l . A l l of these e f f e c t s worsen 19 the detector resolution and often make i t d i f f i c u l t to extract from the spectrum the number of true events associated with the reaction being studied. A block diagram of the elect r o n i c s i s given in Figure .11-2 and a description follows. The signals from the seven phototubes on the Nal(Tl) detector are act i v e l y summed, then sent to a fan-out from which one branch, the l i n e a r signal, i s amplified and sent to a li n e a r gate. A second branch from the fan-out i s cable clipped to a width of 50 nsec, amplified and sent to a constant f r a c t i o n discriminator c a l l e d the High Level Discriminator (HLD). The bias on the discriminator i s set just far enough below the region of interest i n the spectrum that any threshold e f f e c t s of the HLD have disappeared. In t h i s way, pile-up of two low l e v e l pulses i s e f f e c t i v e l y prevented from appearing i n the spectrum. No attempt i s made to discriminate against high-low pile-up. One output of the HLD opens the l i n e a r gate, allowing the l i n e a r s i g n a l to pass to an analog-to-digital converter (ADC) interfaced to the Nuclear Physics Laboratory SDS 930 computer. The s i g n a l i s shaped c o r r e c t l y and delayed appropriately for the ADC by a l i n e a r gate and stretcher. A second output from the HLD i s fed to two coincidence c i r c u i t s via an updating (dead time-less) discriminator to check for coincidences with the AC channel. The signals from a l l eight phototubes on the p l a s t i c s c i n t i l l a t o r s are act i v e l y summed. The resultant pulse i s then amplified, cable clipped to a width of 80 nsec, amplified again and passed to an updating discriminator. The bias l e v e l on FAST AMP TIMING FILTER AMP UPDATE DISC FAST UPDATE . COINC DISC PLASTIC M I (Tl) SHIELD FAN OUT FAST AMP LINEAR AMP CONSTANT FRACTION DISC UPDATE DISC GATE AND DELAY ADC GATE (REJECT) FAST GATE ANTI- AND COINC DELAY ADC GATE (ACCEPT) UPDATE DISC TIMING FILTER \ AMP LINEAR GATE LINEAR GATE STRETCHER F i g . II-2 : Block diagram of the gamma spectrometer e l e c t r o n i c s . o 21 t h i s discriminator i s set just above the noise, somewhere around 100 keV, and the output i s fed into a fa s t coincidence with the output of the HLD {after suitable delays - not shown). k coincidence here implies that energy has been deposited i n both the c e n t r a l c r y s t a l and the surrounding shield. Therefore the l i n e a r signal i s routed into a portion of computer memory lab e l l e d " r e j e c t " , because these events are normally discarded, actually, i n t h i s experiment, the " r e j e c t " spectra were used i n the subseguent analysis. I f there i s no coincidence between the two channels, then i n p r i n c i p l e no energy was l o s t from the central detector. Hence l i n e a r signals for these events are routed into the "accept" portion of memory. This i s accomplished by the second coincidence t e s t , which requires a coincidence between the HLD output and a n u l l output from the f i r s t coincidence (again a f t e r suitable delays). The r e j e c t and accept routing pulses are further s p l i t according to whether the proton beam i s spin up or spin down. This i s accomplished using the f l i p p e r referred to e a r l i e r . In addition, a s i g n a l from a pulse generator was fed d i r e c t l y into the fan-out i n p a r a l l e l with signals from the Nal(Tl) detector. The pulser was f i r e d by the current integrator and hence gave a d i r e c t measure of the dead time i n the detector electronics system. The pulser was also used to check the l i n e a r i t y of the electronics prior to each set of measurements. Various outputs were scaled i n case i t became necessary to consider rejecting a questionable data point. This i s not 22 shown in the figure. .. The scaled outputs included the number of accept and rej e c t routing pulses and the number of counts i n the p l a s t i c s c i n t i l l a t o r s . , In addition, a l l events which deposited more than about 250 keV i n the Nal(Tl) detector were scaled, and the counting rate for these events was kept below 40 kHz. The signals for these low l e v e l events were derived from a separate branch of the fan-out. F i n a l l y , i t was necessary to s t a b i l i z e the phototubes on the Nal (Tl) detector against d r i f t s incurred by variable counting rates. This was accomplished dynamically by an external feedback system which adjusted the high voltage on the tubes i n such a way as to keep constant the height of the pulse from some peak i n the low energy part of the spectrum. Gamma rays from the i n e l a s t i c a l l y excited l e v e l at 4.43 MeV were used for this purpose. An example of a spectrum obtained with the spectrometer i s shown in Figure II-3. The HLD cut off i s noted around E y = 7 MeV, and the part of the spectrum below t h i s energy has been omitted. I t i s seen that the accept spectrum i s considerably improved over the combined spectrum. Removal of those events associated with the loss of one of the pair annihilation guanta i s responsible for most of the improvement. The resolution ( f u l l width at half maximum) of the accept part i s about 4.0% compared to 1,0% for the sum spectrum. The background above the peak res u l t s mainly from cosmic rays, although there i s a small excess over the background expected from t h i s source in the accept spectrum. This excess background i s due partly to high energy gamma rays from the 23 3 0 0 2 0 0 100 co Z O o u . o c_ LU 00 25 Z3 Z 0 2 0 0 PEAK j WINDOW - | SINGLE ESCAPE PEAK HLD CUTOFF 0 • •••• .BACKGROUND WINDOW J 1 I ! 100 0 5 0 JL_ . L e • ACCEPT e • o e e e _L . \ ' V " " e " REJECT j—g.»oooo»o ^ 8 10 12 14 16 18 E/MeV) Fig. I I - 3 : 1 2 C ( p , Y 0 ) 1 3 N gamma ray spectrum at = 11.2 MeV. 24 1*N(p,Y)iso reaction, although the p a r t i c l e y i e l d s indicated there was only a trace of **N i n the target. Some of the excess background might also ari s e from pile-up, or from some unidentified contaminant i n the target. In any event, i t did not prove to be a d i f f i c u l t problem to handle. A peak due to l*0{p, Y Q) * 7 F i s also noted around E y = 11 HeV. Without the excellent detector resolution, t h i s peak would have merged with the 1 2 C (p, Y Q) isjj peak and would have been included in the analysis, although this would not have been a serious problem in the present case. The pulser peak l i e s off-scale at an eguivalent gamma ray energy of about 30 MeV. The window regions i n which the number of counts was summed are also shown. The positioning of the windows i s discussed in Chapter I I I . 2.4 P a r t i c l e Detection Two lithium d r i f t e d s i l i c o n detectors were located in the scattering chamber as shown in Figure II-1. Their purpose was to provide a constant monitor of the beam po l a r i z a t i o n via the *2C(p,p ) * 2C reaction, and to provide a secondary means of data • o normalization. They were symmetrically placed at 160° to the incident beam d i r e c t i o n . This angle was f a r enough back not to inte r f e r e with gamma rays going to the gamma spectrometer when i t was located at back angles, but not so far back to int e r f e r e with the incoming beam. 25 Collimators consisting of v e r t i c a l s l i t s 0.125 inches wide by 0.44 inches high were mounted i n front of the detectors. The distance from the centre of the target to the collimators was 4.25 inches. Signals from the detectors were fed to Ortec 109A preamplifiers located immediately outside the target chamber and thence to the counting room where they were processed. A block diagram of the electronics i s shown in Figure II-4, and a description follows. From the l i n e a r amplifier, one branch was sent to a l i n e a r gate and stretcher, where the signals were delayed appropriately and then passed to a sum amplifier and the ADC., The l o g i c branch was sent to a single channel analyzer (SCA) where a low l e v e l discriminator was used to cut out the low energy pulses. The counting rate f o r pulses above the discriminator threshold was about 2 kHz, The output from the SCA was mixed with the l o g i c signals from the pulser and then sent to route l i n e a r signals present at the ADC into the appropriate portion of memory. The l o g i c signals were also fed to an "exclusive-or" mixer. The purpose of the mixer was to gate the ADC when there was a l o g i c pulse present from only one detector. Otherwise, the ADC would not know from which detector the l i n e a r pulse had come, and i n any event, t h i s l i n e a r s i g n a l from the sum amplifier would probably be a pile-up of pulses from both detectors. The pulser was f i r e d by the current integrator so that dead time corrections could be made d i r e c t l y . Unfortunately, L PART DETECTOPJ BIC" BIC' PREAMP LINEAR AMP L TEST IN OR PULSE GEN R TEST LN R PART PETECTOPJ PREAMP LINEAR GATE . STRETCHER TIMING SCA OR GATE AND DELAY h— SCALER DUAL DECADE ATTEN L TEST OUT R TEST OUT GATE AND DELAY TIMING SCA *\ SCALER OR LINEAR AMP LINEAR GATE STRETCHER GATE AND DELAY SCALER 1 ADC ROUTE SCALER ADC GATE GATE • AND DELAY SCALER ' ADG ROUTE ADC Fig. : Block diagram of the p a r t i c l e detector el e c t r o n i c s to 27 the pulse generator l a t e r appeared to be fa u l t y , and i t was found that the re s u l t s were more self - c o n s i s t e n t i f the counts were not dead time corrected. As with the gamma el e c t r o n i c s , a l l l i n e a r pulses were further separated according to whether the proton spin was up or down. Several of the branches were scaled. These are shown i n the figure. An example of a p a r t i c l e spectrum i s given i n Figure I I - 5 . The strongest peaks are from e l a s t i c scattering o f f 1 2C and i n e l a s t i c scattering leaving 1 2 C i n i t s f i r s t excited state. The peaks r e s u l t i n g from e l a s t i c scattering off l*~ and 1 6 0 are also c l e a r l y seen, but note the logarithmic scale. The pulser peak l i e s below the threshold for l i n e a r signals so that i t i s i n a background-free region of the spectrum.. 2 . 5 Targets Three d i f f e r e n t targets were used in the course of thi s experiment. A natural carbon target of thickness 1.9 mg/cm2 was used for the measurement at E J = 13.5 MeV. i t was found that gamma rays from the 1 3 C ( p , Y ) 1 4 N reaction contaminated the spectrum above the peak of intere s t (natural carbon contains about 1.1% 1 3 C ) . Although i t would always have been necessary to subtract a background due to cosmic rays, the presence of the 1 3 C ( p , Y ) 1 4 N gamma rays made the background subtraction less c e r t a i n . Thus i t was decided to run with pure 1 2 C targets. I t 28 12 C(P»P 4 * / C | .43 CO *-o o y. o 10' PULSER J I0: 1.0* •••• • • • ,2C(p,p0) ,2C , 6 0 C&p o ) , 6 0 I 30 14 K V * N(p,Po)"N .1 A B •* • • • • c • ». • • • • *— •• • • • J «• • I 60 90 120 150 180 CHANNEL NUMBER F i g . II-5 : P a r t i c l e spectrum at E£ = 11.2 HeV. Note the logarithmic scale of the ordinate. The arrows 'A• and » B' define the window region referred to i n the text (Chapter III) . 29 was hoped that only the much more certa i n cosmic ray background subtraction would then be necessary. Accordingly, two targets were ordered from Penn Spectra Tech.* The targets were approximately 1 mg/cm2 thick. The thickness was measured by comparing yields i n t h i s experiment to the e l a s t i c scattering cross section data of Meyer jet a l * (ME 76) and the i n e l a s t i c scattering data of Swint e_t a l . (SW 66). A gamma spectrum from one of the targets i s shown i n Figure II-3 and a p a r t i c l e spectrum i s shown in Figure II-5. Although there i s no evidence for the presence of l 3 c i n these spectra, i t has already been pointed out that there i s some contamination from nitrogen and oxygen., By comparing the p a r t i c l e y i e l d s in t h i s experiment with the d i f f e r e n t i a l cross section data of Daehnick (DA 64), i t was found that the oxygen content in the target was about 0.01 mg/cm2. comparison of the nitrogen y i e l d to the data of Hintz (HI 57) indicates the nitrogen content i s only 0.0004 mg/cm2. A l l runs except those at 13.5 MeV were taken with one or the other of these enriched 1 2 C targets., 2.6 Current Integration The current col l e c t e d in the Faraday cup was measured by a Brookhaven Instruments Corporation (BIC) current integrator. 1Penn Spectra Tech 411 Bickmore Drive Hallingford, Pennsylvania 19086 3 0 The BIC del i v e r s a routing pulse for a certa i n amount of charge co l l e c t e d . These pulses were divided, as usual, according to whether the proton spin was up or down. Scalers were then used to record the integrated charge. The BIC output pulse was also used to f i r e the pulsers i n the detector e l e c t r o n i c c i r c u i t s to keep track of dead times. 2.7 Polarization Measurements The beam polarization was continuously monitored during the runs by the l 2C(p,p ) 1 2 C reaction whose analyzing power i s well known (ME 76). In addition, the polarization was measured several times throughout the runs with a helium polarimeter (BA 75) i n a separate beam l i n e . The p a r t i c l e detectors were placed at 112.5° to the incoming proton beam since at t h i s angle, the analyzing power for the 4He{p,p o)*He reaction i s close to 1.0 f o r a l l the energies measured (SC 71). 2.8 Data Accumulation Three dif f e r e n t runs were made in t h i s experiment. In the f i r s t , data were obtained only for a proton energy of 13.5 MeV. In the second, data were taken at 12 MeV, 14 MeV and 16 MeV. In the f i n a l run, data were gathered at 10 MeV, 11.2 MeV, 12.8 MeV, 15 MeV and 17 MeV. Because the GQR l i e s at a high excitation energy, there should be a large number of allowed decay channels and hence the resonance w i l l be guite broad. 31 Thus i t was f e l t that measurements i n approximately 1 MeV steps would be adequate to survey the region. In order to help detect possible systematic errors, the data at most energies were coll e c t e d with four passes over the angles measured. At one energy (17 MeV), only two passes were made because of time constraints and at two other energies {11.2 MeV and 13.5 MeV) three passes were made. It was necessary to have the face of the target pointing at an angle greater than 20° from the gamma detector angle to avoid absorption through the target holder. Therefore the angles 43°, 55°, 70°, 90°, and 137° were measured with the target at 110°. The angles 43°, 110°, 125°, and 137° were measured with the target at 70°. The end points were measured with the target at both orientations i n each pass to ensure that there were no systematic e f f e c t s associated with the target rotation. None were observed. The angles were chosen to be egual to the zeros of the various Legendre polynomials. There was no other reasonable c r i t e r i o n for the choice of angles - for example, no angle i s more sensi t i v e than another to the presence of E2 radiation. When the gamma spectrometer was located at the forward angles, 6 inches of lead was placed between the beam collimators and spectrometer collimator. This prevented radiation produced by the beam s t r i k i n g the collimators from reaching the Nal(Tl) detector d i r e c t l y . Some runs were taken by c o l l e c t i n g the complete charge of 30 ucoul f i r s t with the proton spin i n one di r e c t i o n , then i n the other. Others were taken with the spin f l i p p i n g 32 automatically once a second. When t h i s was the case, the electronics was automatically shut down for 1 msec while the f i e l d s were reversing. The data for each measurement were stored d i r e c t l y i n the SDS 930 computer. A preliminary analysis of the data was carried out at the end of each run while the detector angle was being changed. The data were also written onto magnetic tape for l a t e r o f f - l i n e analysis. 33 Chapter III DATA ANALYSIS AND RESULTS The main aim of t h i s experiment was to measure a very small E2 cross section in the presence of a very large E1 "background". Thus i t was necessary to s c r u t i n i z e the data very c a r e f u l l y t o ensure that no systematic biasing of the results occurred. In t h i s chapter, the methods used to analyze the data and to check i t s consistency w i l l be described, and then the r e s u l t s of the analysis w i l l be presented. In several places throughout the chapter, comparison i s made between the experimental r e s u l t s and the results of a d i r e c t semi-direct model c a l c u l a t i o n . The model and the parameters used i n the c a l c u l a t i o n s w i l l be described i n Chapter IV. 3.1 Gamma Ray Spectra Analysis There are e s s e n t i a l l y two ways to determine the area of the peaks i n the spectra. One i s to use standard line-shapes to f i t a l l the peaks of i n t e r e s t . , This i s useful when the peaks are s i t t i n g on large backgrounds or when two or more of them overlap. The other i s to define a window around the peak or peaks of inte r e s t and simply sum the counts within t h i s window. In this experiment, the spectra were reasonably clean 34 and the peaks were well separated except for a small number of high energy background gamma rays from proton capture on oxygen and nitrogen. Thus i t was decided to use the second method of analysis, taking care to place the lower l i m i t of the window above the peak from the reaction 1 6 0 ( p , Y Q ) * 7 F , which has a Q-value of 0.6 MeV. I t was also necessary to subtract a small background which arose from the contaminants in the target and from cosmic rays which penetrated the lead shield. The same computer program was used to analyze the data both on-line and o f f - l i n e (BU 75a). A b r i e f description of the analysis procedure follows. F i r s t , a window was defined as a f r a c t i o n (>1.0) above and a f r a c t i o n (<1.0) below the centroid of a strong peak i n the spectrum and an i n i t i a l guess of the centroid of t h i s window was made. A new centroid was then calculated for the window so defined, and from t h i s centroid a new window was defined and a new centroid calculated. This procedure continued u n t i l successively determined centroids agreed to within 0.1 channel, since the error i n the centroid position was t y p i c a l l y 0.1 channel. For the purpose of defining the centroid, the spin up and spin down spectra, including both the accept and reject parts, were summed. When the centroid had been determined, the counts were summed within a second window, also defined as f r a c t i o n s of the centroid. For the data at 16 MeV and 17 MeV, there were strong l i n e s in the spectra from i n e l a s t i c scattering o f f the 12.7 MeV and 15.1 MeV l e v e l s of * 2C, respectively. At these energies, the centroid was determined from these strong peaks since they 35 are less susceptible to s h i f t s due to background variations. For a l l other energies, there was no peak stronger than Y 0 i n the spectra, hence t h i s l i n e was used to define the centroid. An example of a window region defined in t h i s way i s shown i n Figure II-3. The counts within the window for the i n d i v i d u a l spin up and spin down, accept and r e j e c t , spectra, were then summed. Yields in the f r a c t i o n a l channels at the ends of the window were determined by linear extrapolation between the channels above and below the window l i m i t . The counts i n a background window defined above the peak window were summed for each of the four spectra. The background counts were normalized to the number of channels in the peak window, and were then subtracted from each peak sum. The error of the counts i n each peak was given as simply the s t a t i s t i c a l error; that i s , {total area * background area) 1/ 2. The yields were corrected for dead time by dividing by the number of counts in the pulser peak. This automatically corrected for any differences in the charge co l l e c t e d during the spin up and spin down parts of the run. a l t e r n a t i v e l y , the spin up and spin down counts could be normalized according to the number of counts obtained i n the p a r t i c l e detectors. Differences between the normalization methods w i l l be discussed i n section 3. 4. Ratios of counts i n the r e j e c t spectra to counts in the accept spectra were calculated for each spin up and spin down run. In Figure I I I - 1 , these r a t i o s are plotted as a function of angle for one energy. It i s seen that the average values 36 1—3 I hi i 8 0 1 0 0 120 140 "0-|_ab (Degrees) Fig. III-1 : Reject to accept r a t i o s for E£ = 10 MeV. The plus signs are the averages at the given anqles and spin orientations; the surrounding points with error bars are the corresponding experimental measurements. See text for discussion. 37 fluctuate a f a i r amount as a function of angle, p a r t i c u l a r l y for the spin down spectra. In addition, there i s a f a i r f luctuation about the average value at each angle. Although di f f e r e n t i n d e t a i l s , the data at each energy showed similar variations. In some, but not a l l , cases, the fluctuations seemed to be correlated with the t o t a l number of counts recorded i n the AC s h i e l d . I t was because of these variations that i t was necessary to include the r e j e c t spectra i n the analysis. The repeat measurements at each angle were found to be more sel f - c o n s i s t e n t when t h i s was done compared to using only the accept analysis. 3.2 P a r t i c l e Spectra Analysis The p a r t i c l e peaks were also summed within a defined window. Channel locations for the peak window and background windows below and above the peak were read into the computer on cards. An option was to have the program s l i d e the windows u n t i l the centroid calculated for the peak was within one channel of the centre of the peak window. In the i n i t i a l analysis, the four peaks shown in Figure II-5 were analyzed. I t was found in a l l cases that the beam polarization measured by the 1 2 C ( p , P o ) 1 2 C reaction agreed within errors with the measurements using the helium polarimeter. After t h i s was established, the f i n a l analysis was carried out with a broad window defined over the 1 2 C , 1 4N and i*0 e l a s t i c scattering peaks (between arrows * A* and •B* in Figure 11-5} . Summing a l l 38 three peaks improved the s t a t i s t i c a l accuracy and reduced background uncertainties. The program calculated the charge and s o l i d angle asymmetries associated with the polarized beam, and these were monitored throughout the runs. The analyzing power was also calculated and monitored. Expressions for these quantities are given i n Appendix C. , 3.3 Results from the P a r t i c l e Analysis The three asymmetries for the summed peaks at E£ = 15 MeV are plotted i n the upper half of Figure II1-2. These are quite t y p i c a l r e s u l t s . In t h i s particular case, both the s o l i d angle asymmetry and analyzing power show a slow increase as well as more rapid, but not s t a t i s t i c a l l y s i g n i f i c a n t , fluctuations superimposed on t h i s general trend. There i s no simple explanation f o r these results. They could be due to small beam steering effects which may or may not be coupled with target non-uniformities. They could be r e a l beam p o l a r i z a t i o n changes, possibly for only one spin o r i e n t a t i o n . The dashed l i n e s i n the charge r a t i o and s o l i d angle asymmetry plots correspond to measurements made with unpolarized beam. The measurements of the analyzing power with unpolarized beam w i l l be discussed i n section 3.5. It i s in general very d i f f i c u l t to distinguish between beam s h i f t s and polarization changes unless there i s a reaction taking place in the target for which the analyzing power i s 39 0.69 0.68 0.67 1 1 1 1 1 1 1 ANALYZING POWER ASYMMETRIES "~ EjJ = 15 MeV [ I ' 1.15 1.10 1.05 SOLID ANGLE ASYMMETRIES " ~ I* IJ* n f * i l H f f" i ^ - r r 1 1 - - - --II 1 li • -1.05 CHARGE RATIO ASYMMETRIES - J E£= I5 MeV — i t J 1.00 0.95 0.56 0.54 - 1^  ANALYZING POWER — j £ ASYMMETRIES . j * / ' H I E P = I6 MeV 0.52 0.50 S I 1 ! X 1.05 1.00 0.95 SOLID ANGLE ASYMMETRIES 511 f Ep = 16 MeV 5 " i l ' 1 " 1 *** 1 * * 1 1 1 1 1 1 1 , 1 ^ J l l 1 1 ! -" 1 1 1 i r 1 r 5 10 |5 20 25 30 35 RUN NUMBER F i g . TI I-2 : P o l a r i z e d p r o t o n beam a s y m m e t r i e s a t E ± = 15 MeV and 16 MeV. p 40 close to zero. Such i s not the case here, hut the d r i f t s are small in any event. If they are due to polarization changes, then the change of 1 or 2% i s no more than the assumed error in the p o l a r i z a t i o n , as w i l l be seen l a t e r . An exception to these comments occurs in the data for E + •= 16 MeV. Shown in the bottom part of Figure III-2 are the beam polarization and s o l i d angle asymmetries measured at t h i s energy. I t i s seen that the polarization takes a substantial drop at run 18, and then returns to the o r i g i n a l average value i n two stages. There i s no corresponding variation in the s o l i d angle asymmetry, thus there i s f a i r l y strong evidence that the polarization change i s r e a l . The gamma data at 16 MeV were therefore handled s l i g h t l y d i f f e r e n t l y from the data at other energies and t h i s w i l l be discussed i n section 3.5. 3.4 Results from the Gamma Ray Analysis There were four possible f i n a l results for the gamma ray analysis at each angle, according to whether the accept only (ACC) or the accept plus reject (A*R) sums were used, normalized to either the charge collected (Qnorm) or the pa r t i c l e s counted (Pnorm) for each spin orientation. A l l four r e s u l t s were punched out on cards with th e i r respective s t a t i s t i c a l errors and analyzed by a computer program in which the r e s u l t s were averaged at each of the seven angles for a l l four p o s s i b i l i t i e s . The output from this program included these averages and t h e i r respective errors transformed i n t o the 41 centre of mass frame. It was after t h i s averaging was done that i t was noticed that the A + R r e s u l t s were more self-consistent than the ACC re s u l t s . It was also noted that the Qnorm results were more consistent than the Pnorm re s u l t s . , The method of determining these facts was as follows.. At the seven angles measured for each energy, the number of r e s u l t s that were within one standard deviation <1CT) of the average was counted, the number between one and two standard deviations (2a) was counted, etc. Assuming these numbers follow a normal d i s t r i b u t i o n , 67% should be within 1a, and 94.5% should be within 2a. The number of points that could be expected to l i e more than 2a away from the appropriate average can be found from the mean and standard deviation of the binomial d i s t r i b u t i o n where N i s the number of samples and n i s the number of events that occur with p r o b a b i l i t y , p. In the normal case of four passes over the angular d i s t r i b u t i o n , there are 36 data points and, for p = .055, the mean (Np) is 2.0 and the standard deviation (^Np(1-p)) i s 1.4. Thus no more than two or three points would be expected to l i e more than 2a away from the appropriate average. This was always true f o r the Qnorm A+R r e s u l t s . A small increase i n the errors was reguired to make i t true for the Qnorm ACC r e s u l t s . The Pnorm r e s u l t s also needed a s l i g h t increase i n the errors. The l a t t e r r e s u l t can be understood when i t i s r e c a l l e d that the p a r t i c l e y i e l d s were 42 flu c t u a t i n g because of small beam s h i f t s or polarization variations, while the charge collected was not sens i t i v e to these changes. There may also have been small dead time variations in the p a r t i c l e y i e l d s for which no corrections were made. The improvement of the A+R over the ACC r e s u l t s i s understood from the fluctuations mentioned e a r l i e r in the reject/accept r a t i o s . Thus the Qnorm A+R analysis of the raw data was the most self-consistent. In addition, because a larger number of counts was being used in the analysis, the A+S r e s u l t s had smaller s t a t i s t i c a l errors than the ACC r e s u l t s . For the above two reasons, the Qnorm A+R results were used in a l l further analyses. At some energies, the various parameters of i n t e r e s t were extracted using the other three sets of raw data. Disagreement with the Qnorm A+R r e s u l t s occurred only rarely, and no systematic e f f e c t s were observed. An example of an angular d i s t r i b u t i o n and polarized angular d i s t r i b u t i o n obtained for the reaction 1 2 C ( p , Y 0 ) » 3 N i s shown i n Figure III-3. , The ordinate of the angular d i s t r i b u t i o n plot i s the sum of the spin up and spin down yield s . . The ordinate of the polarized angular d i s t r i b u t i o n plot i s the asymmetry, defined as the difference of these y i e l d s divided by 2^A q, where P i s the magnitude of the beam pol a r i z a t i o n , and AD i s related to the t o t a l strength of the reaction. The plus signs are the average values at each angle, the surrounding points with error bars are the actual data. The consistency of the separate measurements i s seen to be very good. 43 Fig. III-3 : The complete angular d i s t r i b u t i o n measurements at E£ = 10 MeV. The ordinate of the upper plot i s the sum of the spin up and spin down y i e l d s . The ordinate of the lower plot i s the difference of these y i e l d s divided by 2<Ph0 (see te x t ) . The plus signs are the averages of the measurements at a given angle; the surrounding points with error bars are the actual measurements at that angle. 44 One other i n t e r e s t i n g test of the consistency of the data sas made. For every energy, the r e s u l t s for the y i e l d and analyzing power at each angle were averaged and a chi-sguare { X 2 ) for the averaging process was calculated. The data were actually averaged in pairs in the order in which they were measured for a given angle ( i . e . the f i r s t two measurements at a given angle were averaged together and then the next two measurements at that angle were averaged) to increase the number of chi-sguare values. The r e s u l t i n g chi-sguares for a l l energies and angles were then counted i n 0.1 wide bins. There were a t o t a l of 128 values for both the y i e l d and analyzinq power. This procedure would be expected to y i e l d chi-sguare d i s t r i b u t i o n s with one degree of freedom. Plotted i n Figure IXI-4 are the r e s u l t i n g histograms - the s o l i d curved l i n e s are the expected r e s u l t s . There appears to be no n o n - s t a t i s t i c a l behaviour i n the analyzing power histogram; there i s possibly a small excess of points between x 2 = 0.7 and 2.1 in the y i e l d histogram. Not shown in t h i s figure are points with x 2 > 5.0. There were 6 of these in the y i e l d and 5 i n the analyzing power. ., The number of chi-sguare values expected to be greater than 5.0 can be found from the binomial d i s t r i b u t i o n as before. The pro b a b i l i t y , p, of x 2 > 5.0 i s .025 for 1 degree of freedom. Then the mean {with N = 128) i s 3.2 and the standard deviation i s 1.8. Thus between 1 and 5 values of chi-sguare are expected to be greater than 5.0. Overall then, there appears to be no strong evidence for n o n - s t a t i s t i c a l behaviour i n any of the data. F i g . III-4 : D i s t r i b u t i o n of x z for the t 2 C ( p , Y 0 ) i 3 N y i e l d s and asymmetries. The chi-sguares were obtained by averaging the data i n pairs (see text). The s o l i d curves represent the expected x 2 - d i s t r i b u t i c n for 1 degree of freedom. 46 3.5 Beam Polarization Measurements The spectra obtained from the measurements using the helium polarimeter were printed out channel by channel and the f i n a l analysis was carried out by hand. The r e s u l t s of these measurements are shown in Table II1-1 along with the polarizations as determined from the 1 2 C ( p V P 0 ) l 2 C reaction. The l a t t e r values are the averages of the runs for the given energy. The analyzing powers f o r the *He measurements were taken from the data of Schwandt et a l . , (SC 71), and for the 1 2C measurements from Meyer et a l . (ME 76) . No p o l a r i z a t i o n r e s u l t s are given for the 10 Me? and 11.2 MeV * 2C(p,p )* 2C o data. Reference to the data of Meyer and Plattner (ME 73b) and T e r r e l l et a l . (TE 68) shows that the analyzing power for 1 2 C (p, p o) *.2C i s varying r a p i d l y at these energies, so that small deviations of the actual beam energy from the measured energy would a f f e c t the analyzing power s i g n i f i c a n t l y . O v e r a l l , the agreement between the two dif f e r e n t measurements i s very good., Since the helium polarimeter measurements were e s s e n t i a l l y free from uncertain backgrounds and the analyzing power i s very close to 1.0 throughout the region, t h e i r average values were used i n the subseguent analysis, and the polarization was assumed to be constant at each energy throughout each s e r i e s of runs. An exception to t h i s was necessary for the 16 MeV data where, as has already been noted, a substantial change i n the polarization occurred. Here the average value of the po l a r i z a t i o n , as measured for each angle throughout the run, Table Summary of Beam Polar i z a t i o n Measurements Reaction Polarization Comments 1 2C(p,P O) 12C .4601.020 P = 13.5 MeV •He(p,p ) o •He .730±.017 before P 12.0 MeV 1 2C(P,P O) 12C .7251.024 EP- 12.0 MeV l 2C(p,p J 12C .7301.020 14.0 MeV 1 2C(P,P o5 » 2 C .7361.018 E P 16.0 MeV *He<prpo) •He .7301.015 after EP = 16.0 MeV 4 He (P,P D) •He .7211.013 before EP = 12.8 MeV i2C ( p , p o ) I 2 C .7301.026 E P 12. 8 MeV *He{p,pQ) •He .737+.010 after EP = 12.8 MeV 1 2C(P,P O) .7201.015 P 15.0 MeV 1 2C(P,P O) -.0011.004 E-> P 15.0 MeV ( c o i l s off) 4He{p,p ) o •He .7311.008 after E> P 15.0 Mev 4He{p,p ) o •He -.0011.007 after E P 15.0 MeV (c o i l s off) 1 2C{p,P O) *2C - EP 10.0 MeV 12C(p,P O) 12C - EP = 11.2 MeV 4He(p,p ) o •He .7391.013 after E? 11.2 MeV 4He(p,p ) 0 •He .0121.006 after E-> P = 11.2 MeV ( c o i l s off) 1 2C(p,p ) o 12C .7101.030 E P 17.0 MeV HQ was used. Also shown i n Table III-1 are the r e s u l t s of measuring the pol a r i z a t i o n with unpolarized beam. These are i n d e n t i f i e d as " c o i l s o f f " . The purpose of these measurements was to establish whether or not there were any indications of systematic e f f e c t s contributing to the asymmetries with polarized beam. It can be seen that any deviations frcm zero are usually small and i n s i g n i f i c a n t . 3.6 Angular D i s t r i b u t i o n Functions The angular d i s t r i b u t i o n of gamma radiation following the capture of unpolarized p r o j e c t i l e s has been developed by several authors (for example, BI 60, HO 67, BL 52), and i s given by LW u ( 9 )> ^ C(t,t',k) Re(R tR*,) P k(cos 9) III-2 t,t' ,k where Rt,Rt, a r e reduced reaction matrix elements (T-matrix elements) corresponding to different channels t,t» C(t,t*,k) represents a sum over angular momentum coupling c o e f f i c i e n t s and P {cos e) are Legendre polynomials. The maximum value of k i s given by well known theorems l i m i t i n g the complexity of angular d i s t r i b u t i o n s . Experimentally, the measured angular d i s t r i b u t i o n can be represented by 49 w u.(e) ^ ^ ^ ( c o s e ) i n - 3 k where the Q, correct for the f i n i t e size of the detector, and k are given by Rose (RO 53) . Thus the experimentally determined c o e f f i c i e n t s . A, , can be related to the T-matrix elements through the c o e f f i c i e n t s C ( t , t ' , k ) , as follows, A kQ k = Y. C(t,.t',k) R e ( R t R * , ) I I I-4 t , t ' Methods for c a l c u l a t i n g these c o e f f i c i e n t s have been given by Sharp et a l . (SH 54) among others. I t i s shown i n the work of Devons and Goldfarb (DE 57), following the development by Satchler (SA 55), that for the case of p a r t i a l l y polarized spin 1/2 p a r t i c l e s , the expression 111-2 must be modified by making the replacement Re(R tR*,) P k(cos 9) — Re(R R * t ) P k(cos 9) + I m ^ R * , ) f k ( t , t ' ) fi-fi P*(cos 9) Here P i s the incident beam pol a r i z a t i o n , n i s a unit vector normal to the reaction plane (in the d i r e c t i o n defined by the Madison Convention (BA 7 0 ) ) , the P k(cos e ) are associated Legendre functions, and the factor f (t,t*) i s given by* k 1 Snover and Ebisawa (SN 75) have found that f,{t,t*) d i f f e r s by an o v e r a l l sign from that given by Devons and Goldfarb. , 50 f f t t n -i'Ct'+i) + I(i+D - .1C.1+D - rCi'+D I I I _ 5 V * ' ' ; " k(k+i) where j and j ' are t o t a l angular momentum quantum numbers of the incident p r o j e c t i l e correspondinq to o r b i t a l angular momenta i and V, respectively. Thus, equation I.II-2 becomes W ( 8 ) ^ C(t,t',k)[Re(R tR* t) P k(cos 6) + t , t \ k III-6 Im-(R R j) f k ( t , t ' ) /P-n P k(cos 6) For the case in which the proton spin i s perpendicular to the reaction plane, the results of measuring the angular d i s t r i b u t i o n of the gamma rays can be expressed as the sum and difference of the yields obtained with the proton spin up {•»+) and spin down . The sum gives the f a m i l i a r unpolarized angular d i s t r i b u t i o n , ®uiQ)t where k and the difference can be expressed in terms of the analyzing power A (0) , as W u(9)A(6) * W t ( 9 ) 2 - / + ( 6 ) = I B kQ kP k(cos G) III-8 Comparison of III-7 and III-8 to II.1-6 shows that the experimentally measured A can be related to the T-matrix elements by the angular momentum coupling c o e f f i c i e n t s C(t,t 1,k) as before, but now, with a polarized beam, the new experimentally determined guantities B can also be related to 51 the T-matrix elements through the simple m u l t i p l i c a t i v e factor f j t , t ' ) as follows, \Q k = Ic(t,t',k) Im(RtR*,) f k(t,t') III-9 t,t' It i s convenient to factor out A , which i s a measure of o the o v e r a l l strength of the reaction, from the expressions above. Then III-7 and III-8 become Wi(9) + W4-(9) 2 = A Q [ l + £a kQ kP k(cos 6)] 111-10 Variations in a- and b with energy are caused only by variations i n the T-matrix elements, and not by changes in the o v e r a l l strength. The appropriate T-matrix elements are determined by noting the p a r t i a l waves which take part in the reaction. For the case of spin 1/2 p a r t i c l e s incident on a spin 0 nucleus leading to a f i n a l state with spin 1/2 and a gamma ray, as i s the case here, only four p a r t i a l waves can contribute, i f the radiation i s r e s t r i c t e d to being only El and E2. By angular momentum and parity conservation, these are, in j j coupling, s and d . capture which lead to E1 radiation and p , and f , capture 3/2 5/2 which lead to E2 r a d i a t i o n . Thus the T-matrix elements can be i<j>„ !((>„• X<J>J i<j>f labelled se , pe , de , and fe , where s and <j>s are the amplitude and phase for the s 1 ^ 2 P a r t i a l wave, etc. The great advantage i n using polarized beams now becomes 52 apparent. There are seven T-matrix element parameters to be determined - four amplitudes and three r e l a t i v e phases. If the highest multipolarity of Y-radiation i s two then the maximum value of k i n the summations III-7 and III-8 i s four. Hence with unpolarized beam, only f i v e c o e f f i c i e n t s can be measured experimentally, but with polarized beam, nine c o e f f i c i e n t s can be measured. Thus measurment with a polarized beam enables, i n p r i n c i p l e , the determination of a l l seven T-matrix elements. The r e l a t i o n between the A, and B c o e f f i c i e n t s and the k k T-matrix parameters are l i s t e d i n Table III-2. The capture amplitudes have been renormalized so that the sum of the squares of the amplitudes w i l l be egual to A q . The r e l a t i o n between these reaction matrix elements and the actual reduced matrix elements (R^R^,) i s given in Appendix B. It i s seen from t h i s table that A2 i s dominated by e l e c t r i c dipole terms, with only incoherent contributions from the much weaker guadrupole terms. The c o e f f i c i e n t A^ i s a pure e l e c t r i c guadrupole term and can therefore be expected to be very small, while Aa and A 3 result from dipole-guadrupole interference. The same considerations hold f o r the B, 1 s, k although, as shown in Table III-2, these are pure interference terms {i.e., no terms such as s 2 , p 2, d 2 , or f 2 are present) and B 2 w i l l not necessarily be dominated by dipole radiation i f the s,d phase difference i s near 0° or 180°. Thus i t i s seen that the dominant effect of the presence of guadrupole radiation i s i t s interference with the dipole r a d i a t i o n , and the consequent appearance of Legendre and associated Legendre functions of odd degree., 53 Table III-2 Relation between the Angular D i s t r i b u t i o n C o e f f i c i e n t s and the Reaction Matrix Elements * A = s 2 * p2 + d 2 • f 2 o A. = 2.45spcos(d> -<f> ) - 0. 35pdcos(<j> -<j>.J + 2 .55dfcos (<j> - <f> ) 1 s p p d d f-A0 = 0.5p2 - 0.5d2 + 0.57fz * 1.41sdcos ((f> -<f> ) 2 s d - 0. 35pf cos (4> - ( f ) ) P f A Q = 2.00sfcos{d, - d i . ) * 2.08pdcos(<f> -<j> ) - 1.13dfcos (d> -<f> ) 3 s r p d d f kh = -0.57f 2 + 2.80pfcos ( < f > p - ( f ) f ) B : = 1.22spsin ( < | ) s - < f ) p ) + 0.69pdsin(* p - ( f > d ) - 1. 27df sin ( * d - < l > f ) B 2 = -0.71sdsin< < l ' s - ( t ' d ) + 0.2 9p.f sin ( < f > p - < ! > f ) B 3 = -0.67sfsin ( < ( ' s - < f ' f ) - 0. 69pdsin ( < f > p - < f > d ) + 0. 09df s i n ( ( f > d - < f > f ) B^ = -0.70pfsin (<f> p -<f> d ) * The r e l a t i o n between the reaction matrix elements and the R of eguation III-2 i s given in Appendix B., The s o l i d angle correction factors Q k are l i s t e d i n Table III-3. They were calculated for the p a r t i c u l a r detector arrangement described in Chapter II using a computer program written for t h i s purpose (LE 64)., 54 Table III-3 S o l i d Angle Correction Factors Q0 Q l Q2 V ®h 1.000 .995 .985 .970 .950 3.7 Results of the Angular Distribution F i t s The averaged data at each energy were f i t by a linear least sguares technique with the angular d i s t r i b u t i o n functions III-7 and III-8.. This f i t was performed as the f i r s t step of a two part computer analysis (BO 76a). Since the eguations involved are l i n e a r i n the parameters, this c a l c u l a t i o n i s quite straightforward and closely follows the prescription given by Bevington (BE 69a). The d i f f e r e n t i a l cross sections and f i t s are plotted in Figure III-5, and the asymmetries and f i t s are plotted i n Figure III-6. The extracted c o e f f i c i e n t s and their errors are l i s t e d i n Table III-4. Note that the chi-sguares that are guoted have been divided by the number of degrees of freedom, v, and are thus reduced chi-sguares (x 2 , ~ x 2 / v ) • I t i s seen that a l l these f i t s are acceptable i n the sense that the chi-sguares are reasonable. The worst case for the asymmetry occurs at 13.5 MeV, where the x 2, of 1.87 corresponds to a 13% confidence l e v e l f o r 3 degrees of freedom, which i s c e r t a i n l y acceptable. The worst case for the yield occurs at 14 MeV; there the reduced chi-sguare of 4.24 corresponds to a 1.4% confidence l e v e l f o r 2 degrees of freedom. While t h i s i s only marginally acceptable, nothing unusual was noted in the further 55 i ~i i I | — — i 1 r J i i I I L i i 45 90 135 45 90 135 6. A T } (DEGREES) Fig. III-5 : 1 2 C (p, Y D ) » 3 N normalized d i f f e r e n t i a l cross sections. The s o l i d l i n e s are from a least sguares f i t to the data (see t e x t ) . S t a t i s t i c a l errors are shown where they are larger than the spot s i z e . 56 i 1 1 — 1 I 1 1 r J I L_ I I I L L 45 90 135 45 90 135 e T A R (DEGREES) F i g . III-6 : * 2 C ( p , Y 0 ) i 3 N angular d i s t r i b u t i o n s for the asymmetries. The s o l i d l i n es are least sguares f i t s to the data (see t e x t ) . S t a t i s t i c a l errors only are shown. Table III-4: C(p,y ) N Angular Distribution Coefficients P 4TTA o a l a2 a3 a4 ' b l b 2 b 3 b4 2 X v — (MeV) (ub) (c) (a) (b) 10.0 22.0 .120 ±.012 -.384 ±.039 -.100 ±.028 -.047 ±.045 .0118 ±.0088 .1916 ±.0055 .0248 ±.0053 .0018 ±.0064 1.02 1.54 11.2 21.0 .086 ±.012 -.620 ±.041 -.114 ±.029 .081 ±.045 -.0711 ±.0097 .1125 ±.0053 .0267 ±.0058 .0124 ±.0066 1.02 0.49 12.0 24.9 .218 ±.011 -.804 ±.038 -.118 ±.024 -.032 ±.041 -.0265 ±.0082 .1602 ±.0047 .0426 ±.0050 -.0009 ±.0056 0.66 0.90 12.8 26.4 .147 ±.010 -.641 ±.034 -.186 ±.024 .100 ±.036 -.1140 ±.0078 .1358 ±.0044 .0418 ±.0047 .0096 ±.0054 3.58 1.15 13.5 22.7 .167 ±.030 -.525 ±.092 -.106 ±.085 .104 ±.085 -.0571 ±.0224 .1860 ±.0179 .0334 ±.0165 -.0077 ±.0190 1.04 1.87 14.0 20.8 .270 ±.014 -.766 ±.050 -.134 • ±.034 . -.052 ±.055. -.0871 ±.0117 .1713 ±.0064 .0498 ±.0069 .0011 ±.0079 4.24 0.51 15.0 17.8 .197 ±.014 -.713 ±.048 -.269 ±.033 -.023 ±.052 -.0341 ±.0109 .1023 ±.0059 .0394 ±.0064 '.0182 ±.0074 1.15 0.49 16.0 15.7 .245 ±.017 -.497 ±.055 -.148 ±.033 .018 ±.062 -.0359 ±.0132 .1522 ±.0076 .0538 ±.0080 .0186 ±.0092 0.93 0.13 17.0 10.7 .253 ±.037 -.230 ±.107 -.212 ±.077 -.045 ±.121 .0275 ±.0241 .1551 ±.0157 .0251 ±.0157 .0272 ±.0177 2.89 0.24 (a) Reduced chi-squared for f i t to yield angular distribution (b) Reduced chi-squared for f i t to asymmetry angular distribution (c) A Q taken from Figure 6 of reference BE 76b U l 58 analysis of the 14 MeV data. Plots of the a. and b c o e f f i c i e n t s are given in Figure k • k III-7. Also shown in thi s f i g u r e are the unpolarized angular d i s t r i b u t i o n c o e f f i c i e n t s obtained by Berghofer et a l . (BE 76b). Only the a and a 2 c o e f f i c i e n t s are compared, because when the f i t s to the data of Berghofer et al.were extended to include a 3 and a^ terms, the errors in a x and a 2 increased to such an extent that the ov e r a l l agreement with the present r e s u l t s was obscured. The s o l i d and dotted l i n e s are the r e s u l t s of c a l c u l a t i o n s with the DSD model, which w i l l be discussed i n Chapter IV. It can be noted here that the presence of the non-zero a 3 and b 3 c o e f f i c i e n t s throughout th i s energy region unambiguously implies e l e c t r i c dipole-guadrupole interference. Both a x and bj are also seen to be non-zero. This could a r i s e from E2 radi a t i o n , but i t can also result from the presence of M1 radiation (see Appendix B). This problem i s dealt with in more d e t a i l l a t e r . 3.8 Extraction of the T-matrix Elements The capture amplitudes and phases were determined from the extracted A, and B, c o e f f i c i e n t s as the second step in the k k computer analysis. In t h i s part of the analysis, use i s made of the gradient expansion algorithm of Marquardt (MA 63) to perform a non-linear l e a s t squares f i t to the eguations of Table III-2. The f u l l error matrix i s retained from the f i r s t 59 -.1 -.2 -.3 +.r o i +.! 0 -.1 -.2: +.1 0 +.05' - . 0 5 - A^Ao I. L - " B / A 0 n r I T L B / A C a / A 0 £ f =5= 10 II 12 13 14 15 4° b(MeV) 16 T i 17 Fig. I I I - 7 1 2 C ( p , Y 0 ) » 3 N normalized angular d i s t r i b u t i o n c o e f f i c i e n t s . Solid points refer to the present data; open c i r c l e s refer to the data of reference (BE 76b). The s o l i d and dotted l i n e s are from c a l c u l a t i o n s with the DSD model. 60 part of the analysis, including the correlation terms among a l l the fl^ and c o e f f i c i e n t s . I t has been found in previous analyses that these co r r e l a t i o n s i n some cases a f f e c t the values obtained for the T-matrix elements, and always affect t h e i r uncertainties (BU 76a). Two solutions with acceptable chi-sguares are found at each energy, one corresponding to dominant d-wave capture (solution I ) , and the other corresponding to dominant s-wave capture {solution I I ) . The extracted reaction amplitudes and phases and associated errors for the two solutions are l i s t e d i n Tables III—5 and III-6, along with the values of the reduced chi-sguare f o r each f i t . Note that both solu t i o n I and solution II occur at exactly the same value of x * 1 ^ n addition, although i t i s not shown e x p l i c i t l y in these tables, both solutions occur at exactly the same value for the E2 cross section ( a E 2 ) . Most of the reduced chi-sguares are c l e a r l y acceptable; the only possible exception i s fo r the f i t at 12.8 MeV where the value of 4.21 corresponds to a confidence l e v e l of 1.5% for 2 degrees of freedom. The amplitudes and r e l a t i v e phase for E1 capture are plotted in Figure III-8. The s o l i d l i n e s in the upper part of the figure correspond to the d-wave and s-wave amplitudes, respectively, calculated with the DSD model and the s o l i d l i n e i n , the bottom part of the figure corresponds to the phase difference between the s-wave and d-wave, also calculated by the model. The calculated d-wave amplitude i s seen to agree well with the d-wave amplitude from solution I. E s s e n t i a l l y 6 1 Table III-5 " C ( p , Y o ) i Solution I. T-matrix Element F i t s to * 2C(p, Y * 3N Angular D i s t r i b u t i o n s . E-> s P a f <j> -<j> <f> ~<f> p (MeV) 2 10.0 .292 .040 .9546 .041 -66° 74.5° 65° 0.54 ±.008 ±.015 +.0035 ±.006 ±17° ±2.2° ±12° 11.2 .240 .072 .9666 .056 -28° 134.7° 54<> 2.85 ±.013 +.022 ±.0048 ±.010 ± 9° ±2.8° ±12° 12.0 .369 .058 .9236 .087 68° 138.2° 157° 0.87 ±.016 ±.012 ±.0078 +.016 ±25° ±1.8° ± 9° 12.8 .321 .129 .9322 .106 -31° 137.5° 51° 4.21 ±.015 ±.023 ±.0087 ±.012 ± 4° ±2.0° ± 7° 13.5 .292 .042 .9533 .068 -35° 103.8° 68° 0.17 ±.028 ±.038 ±.0108 ±.025 ±46° ±5.0° ±24° 14.0 .344 .030 .9314 .116 41° 131.2° 106° 0.43 ±.017 ±.014 ±.0071 ±.015 ±59° ±2.6° ± 9° 15.0 .238 .115 .9635 .041 16° 140.0° 125° 2.69 ±.015 ±.024 ±.0057 ±.031 ± 7° ±3.0° ±17° 16.0 .232 .066 .9671 .081 -12° 101.2° 101° 1.61 ±.012 ±.016 ±.0046 +.012 ±20° ±4.3° ±10° 17.0 .282 .067 .9514 .104 -117° 53.6° 59° 1.31 ±.026 ±.042 ±.0104 ±.017 ±21° ±6.0° ±12° a l l t h e o r e t i c a l models predict that the dominant t r a n s i t i o n i n the GDR w i l l be the one in which the o r b i t a l angular momentum of the absorbing p a r t i c l e i s increased by one unit and there i s no spin f l i p (WI 56). In the present case, t h i s corresponds to 62 Table III-6 T-matrix Element F i t s to l 2 C ( p , YQ) * 3 j j angular Distributions, Solution I I . E-> s p d f $ -<f> <t>c-<\> x2 p p s d s f s v (MeV) 10.0 .931 .035 .361 .045 13° 126.8° 142° 0.54 ±.004 ±.008 ±.009 ±.014 ±16° ±1.6* ±14° 11.2 .857 .059 .507 .070 91° 157.7° 168° 2.85 ±.009 ±.010 ±.013 ±.019 ±15° ±1.0° ±10° 12.0 .783 .085 .612 .061 -17® 151.8° 88° 0.87 ±.013 ±.020 ±.015 ±.014 ± 8° ±0.8° ±21° 12.8 .803 .111 .572 .125 95° 153.9° 173° 4.21 ±.013 ±.013 ±.014 ±.021 ± 9° ±0.8° ± 5° 13.5 .881 .065 .467 .046 37° 139.0° 150° 0.17 ±.017 ±.024 ±.026 +.034 ±27° ±3.3° ±45° 14.0 .807 .117 .578 .029 22° 148.9° 132° 0.43 ±.013 ±.018 ±.017 ±.022 ±10° ±1.2° ±44° 15.0 .849 .027 .514 .119 44° 160.3° 155° 2.69 ±.010 ±.012 ±.015 ±.021 ±36° ±1.1° ± 7° 16.0 .900 .075 .423 .072 43° 144.7° 121° 1.61 ±.008 ±.015 ±.015 ±.017 ±10° ±1.8° ±16° 17.0 .956 .089 .267 .086 -9° 122.1° 167° 1.31 ±.011 ±.024 ±.025 ±.039 ±13° ±6.5° ±16° dominant d-wave capture, and so agrees with solution I. Therefore, most of the future discussion w i l l be based on t h i s solution. The r e l a t i v e phase <|>d- <f>g shows some very i n t e r e s t i n g 63 1.0 0 . 5 1.0 0 . 5 2 0 0 c ! 0 0 c o s • 6 » • « » ° J5-S0LUTI0N I o « o o C Q SOLUTION 3T o o o o o s •d-wave o s-wave •d-wave os-wave s 0(T0s _J L ©Solution I ©Solution IE 10 II 12 13 14 15 16 17 EJ:Q b (MeV) F i g . I I I - 8 E 1 amplitudes and r e l a t i v e phase. Errors are shown where they are larqer than the point s i z e . The s o l i d and dotted curves represent c a l c u l a t i o n s with the DSD model (Chapter IV) . 64 structure. There appears to be a broad o v e r a l l resonance to t h i s phase difference, and a substantial dip near 13.5 HeV. The r e l a t i v e change i n the phase possibly indicates that only one of the reaction amplitudes participates i n the pygmy resonance. The dip might r e s u l t from interference between the pygmy and some l e v e l near 13.5 MeV. The nearest candidate i s the l e v e l observed by Hasinoff et a l . (HA 72) at E = 14.04 MeV — — x i n i 3 N (E+ = 13.12 MeV) with a width of ^ 170 keV. I t would be P necessary to measure in f i n e r energy steps to c l a r i f y t h i s point. Parameters associated with the E2 reaction matrix elements for solution I are l i s t e d in Table III—7, and are plotted i n Figure III-9 along with the res u l t s of the DSD c a l c u l a t i o n . The approximate constancy of the p,d phase difference i s very i n t e r e s t i n g i n that i t implies that i t i s the s-wave phase that i s resonating, unless the p-wave and d-wave phases happen to both be undergoing the same phase changes, which would be very surprising. There appears to be some fluctuations i n the f,d phase difference between E^ = 10 MeV and 14 MeV. Here again i t would inter e s t i n g to measure i n f i n e r steps to investigate t h i s structure more f u l l y . The errors quoted for a l l of the parameters extracted from the data are s t a t i s t i c a l errors only, and do not include such possible errors as beam s h i f t s on the target or errors in the charge c o l l e c t i o n and beam p o l a r i z a t i o n measurements. I t has already been shown that the data i s self - c o n s i s t e n t without including errors from these sources., The eff e c t of varying the polarization by 2% caused about a 4% change in the B v 65 Table III-7 E1 Amplitude Ratio s/d and Parameters Related to E2 Capture for Solution I E+ P (MeV) s/d p/f +p-*f ^ f ' ^ j ^p'^d 'E2 aEl+ 0E2 10.0 .306 .99 -131° - 9° ±.010 ±.49 ±13° ±12° -140° .00328 ±17° ±.00081 11.2 .248 1.28 - 82° -81° ±.014 ±.41 ±11° ±12° -163° .0083 ± 9° ±.0035 12.0 .399 .67 - 97° 19° ±.020 ±.15 ±25° ± 8° • 78° .0109 ±25° ±.0037 12.8 .344 1. 22 - 82° - 87<> ±.020 +.20 ±7«> ± 8 ° •169° .0279 ± 50 ±.0076 13.5 .306 .61 -103° -36° -139° .0063 ±.033 ±.73 ±29° ±24° ±47° ±.0025 14.0 .369 .26 - 65° -25° - 90° .0144 ±.021 ±.14 ±52° ± 9° ±59° +.0030 15.0 .247 2.8 -142° -15° -157° .0149 +.017 ±1.4 ±13° ±17° ± 7° +.0045 16.0 .240 + .013 .81 ±.30 •114° ±17° 0° ± 9° •114° ±20° .0109 ±.0012 17.0 .296 .64 -176° 5° -171° .0153 ±.030 ±.50 ±21° +11° ±20° ±.0036 c o e f f i c i e n t s , but t h i s usually resulted i n a change of less than 1% in the values of the T-matrix elements. Because the eguations of Table III-2 are non-linear, there i s no guarantee that ad d i t i o n a l solutions do not exi s t . An F i g . III-9 : The amplitude r a t i o and phases related to E2 capture. The s o l i d and dotted curves represent c a l c u l a t i o n s with the DSD model (Chapter IV). 67 attempt to locate at lea s t some of these other solutions was made i n the following way. F i r s t , a E 2 was fixed at some arbi t r a r y value expressed as a fraction of the t o t a l cross section, and a l l the other parameters were allowed to vary to minimize x 2 • Then a was stepped to a new value and the E2 process was repeated, with the star t i n g guesses f o r the parameters at each successive step being the values obtained i n the previous step. In t h i s way, the projection of the multi-dimensional x 2 ~ s u r f a c e was cast onto the E 2 strength axis. The results of this search are plotted in Figure III-10. These results were obtained with solution I as the s t a r t i n g point. The f i r s t , and deepest, minimum i s i n each case the doubly degenerate solution corresponding to solutions I and II described previously. The second minimum i n each case appears to correspond to a solution which has a diff e r e n t value of the s/d r a t i o than i s obtained for the solutions at the f i r s t minimum. A t y p i c a l value of s/d for solution I i s 0.3, while a t y p i c a l value for the second solution i s 0.7. The parameters that are obtained at the second minima are l i s t e d i n Table I I I - 8 . At some, but not a l l , energies, the second solution was also found to be doubly degenerate. These other second solutions, where they appeared, were found to belong to the family that begins with solution I I . No exhaustive search was made to fin d them a l l . Several of the second solutions can be excluded on s t a t i s t i c a l grounds. Shown i n Figure III-10 i s the 1% confidence l i m i t f or each f i t ( x 2 •= 9 . 2 ) . A l l of the solutions 68 1 I 1 I 1 I 1 I 1 I 1 I 1 I I I ' I 1 I 1 I 1 I 1 I 1 I 1 I 10 MeV 1 I l I l 1 I I l 1 l I i I i I 1 I • I i I . I . I . I . I . I 0 0.08 0.16 0.24 0 0.08 0.16 0.24 °E2 TOTAL F i g . I l l - 1 0 : Projection of the multidimensional x 2-surface onto the E2 strength axis. 69 Table I I I - 8 Second Solutions to the T-matrix Element F i t s E^ P {MeV) s/d p/f p T f V f *d p d JE2 CTEl+aE2 10.0 • ±. 790 022 * ±. 73 3 026 135.8° ±1.4° -179° ± 7° -67.6° ± 2.6° 113° ± 5° • ±. 1509 0053 24.7 11.2 * ±. 647 029 • ±. 719 068 154.7° ±1.3° - 133° ±13° -91.1° ± 5. 1° 136° ± 8° • ±. 0849 0080 25.5 12.0 • 642 035 1. ±. 11 16 148.2° ±1.2° -167° ±21° 72.8° ± 9. 4° -95° ±12° * ±. 0402 0049 5.22 12.8* 13.5 # ±. 710 089 • ±. 560 059 140.4° ±4. 8° -202° ±38° -59. 1° ±10.5° 99° ±27° * ±. 09 32 0194 3. 79 14.o • 659 038 * 391 084 145.7° ±1.5° -186° ±32° -56.2° ± 7.2° 118° ±25° * ±. 0562 0071 2.43 15.0 • 767 034 « ±. 806 088 158.8° ±1. 3° -143° ±13° -80.0° ± 5.7° 137° ± 8° • ±. 0902 0078 13.3 16.0 • ±. 833 040 • 727 07 2 146.1° ±1.8° -146° ±12° -75.7° ± 4.9° 138° ± 8° • ±. 1234 0089 11.6 17.0 • 787 063 * ±. 941 102 136.5° ±4.8° - 151° ±14° -72.9° ± 5.5° 137° ± 9° ±. 2165 0172 5.97 * No second solution could be found at 12.8 MeV {see text). at the lowest value of o E 2 f a l l below th i s l i m i t , but of the second solutions, only those at 12 MeV, 13.5 MeV, 14 MeV, and 17 MeV f a l l below the l i m i t . Reference to Table III-8 shows that the second solutions at 15 MeV and 16 MeV can only just be excluded on t h i s basis. No second solution could be found f o r the data at 12.8 MeV. Searches s t a r t i n g from many d i f f e r e n t parameter sets were 70 made, but they always resulted in convergence either to solution I or to solution I I . ,. A few other solutions were found which also had E2 cross sections that were larger than the solution I values. The minima of chi-sguare for these additional solutions always lay at unacceptably high l e v e l s . The extraction of a l l the solutions described thus far are subject to the condition that there i s no M1 radiation involved in the reaction. I t was pointed out by Hanna et a l . (HA 74a) that a consistency check on t h i s condition can be made by excluding Aj and from the analysis. These c o e f f i c i e n t s are the most sensi t i v e to the presence of M1 radiation (see Appendix B). using the T-matrix elements r e s u l t i n g from such an analysis, the A2 and Bl c o e f f i c i e n t s can be calculated and compared to the experimental values. In t h e i r analysis of the 1 SN (p, yQ) 16o reaction, Hanna et a l . found that these c o e f f i c i e n t s were s a t i s f a c t o r i l y reproduced from an analysis which excluded them. The results of such an analysis for the present data are plotted in Figure III-11. It i s seen that the calculated c o e f f i c i e n t s do not agree with the experimental c o e f f i c i e n t s i f only the experimental errors are taken into account. I f the errors i n the calculated c o e f f i c i e n t s are also taken into account, then the agreement i s much more sa t i s f a c t o r y . Several of the points l i e more than two combined standard deviations apart, however, and t h i s could be taken to be an indication that there i s some M1 radiatio n underlying the structure i n t h i s region. 71 Et ° b (MeV) F i g . 111-11 The normalized kl and B1 angular d i s t r i b u t i o n c o e f f i c i e n t s . S o l i d points correspond to angular d i s t r i b u t i o n f i t s to the experimental data; open c i r c l e s correspond generated from a other seven to and B l c o e f f i c i e n t s A, and B, k k T-matrix element f i t to the c o e f f i c i e n t s . 72 Two of these offending points l i e at incident proton energies of 11.2 MeV and 12.8 MeV, which are near the interference dips i n the 90° y i e l d curve observed by Measday et a l . (ME 73a). It seems to be f a i r l y well established that the upper of these dips i s caused by a 3/2 + l e v e l i n t e r f e r i n g with the broad pygmy resonance. There are s t i l l some ambiguities about the lower dip, however, and there has been some indicatio n of M1 strength i n t h i s region. Fleming jet a l . (PL 68) observed levels in l 3N at ex c i t a t i o n energies of 10.78 and 11.88 MeV. The decay of the mirror l e v e l i n l 3 C corresponding to the 11.88 MeV l e v e l has been observed to be consistent with M1 radiation i n an i n e l a s t i c scattering experiment (WI 69). In the present experiment, t h i s l e v e l would be excited at an incident proton energy of 10.76 MeV. The width of t h i s l e v e l i s 130 keV (A3 70) , and i t s spin-parity assignment i s 3/2" (HS 71). I f t h i s l e v e l indeed decays by M1 radiation, the large discrepancy observed at E^> = 11.2 MeV between the experimental k1 c o e f f i c i e n t and the Aj c o e f f i c i e n t reproduced from the analysis i n which i t was excluded could possibly be explained. The decay of the 10.78 MeV l e v e l i s also consistent with H1 (or possibly E2) r a d i a t i o n (91-69). This l e v e l would be excited at an incident proton energy of 9.58 MeV in the present experiment. The dominant M1 strength in t h i s region occurs in the decay of the f i r s t T = 3/2 state in at E = 15.07 MeV J X {DI 68). However, t h i s state i s very narrow, so i t s effects 73 should not be f e l t more than a few tens of k i l o v o l t s on either side of the resonance energy {E+ = 14.23 MeV). Of course, i t i s also possible that more solutions e x i s t for the analysis with k1 and 3^ excluded. Only at one energy was a second solution found, however, although several d i f f e r e n t starting guesses were used at each energy. There are three problems with the analysis in which A.j and B 1 are excluded. The f i r s t i s that the minima i n x 2 _ s P a c e are very narrow so the solutions are d i f f i c u l t to f i n d , which leads to the problem just mentioned. The second i s that because seven T-matrix elements are being f i t t e d to seven experimental guantities, there are 0 degrees of freedom in the f i t , so x 2 must in fact vanish at the solution. Thus i t i s not possible to judge whether a solution i s acceptable based on a chi-sguare c r i t e r i o n . F i n a l l y , exclusion of Ax and B2 puts the onus of providing evidence for the existence of E2 strength more on the other c o e f f i c i e n t s . Conseguently, the errors on the extracted E2 cross sections are much larger than when and B^ are included in the analysis. 3.9 Determination of the Cross Sections It i s seen from the above discussion that i t i s possible to obtain at least three solutions to the parameter of central i n t e r e s t - the E2 cross section. The re s u l t s are plotted i n Figure 111-12. The energies at which there are two s t a t i s t i c a l l y acceptable solutions {at a ^% confidence limit) 7a < r E 2 F i g . ITI-12 The E2 cross sections. The s o l i d points are from solution I (or II) and a d d i t i o n a l solutions which s a t i s f y a 1 .0 % confidence l i m i t (see te x t ) . Open c i r c l e s are from solutions obtained with Aj and excluded from the T-matrix element f i t . The s o l i d and dotted curves are from a DSD capture model c a l c u l a t i o n (Chapter I V ) . 75 when a l l the c o e f f i c i e n t s are included i n the analysis are indicated by two s o l i d points at that energy on the graph. The open points correspond to solutions obtained with Aj and B1 excluded from the analysis. The values of a E 2 were calculated by comparing the f r a c t i o n of E2 strength found in the present analysis to the t o t a l cross section results of Berghofer et a l . (BE 76b) . The errors shown do not include the o v e r a l l ±20% normalization error from t h e i r analysis. (Because of time constraints, i t was not possible to measure the e f f i c i e n c y of the detector; i . e . the r a t i o of the number of events recorded i n the analysis window to the t o t a l number of events i n i t i a t e d i n the detector by Y-rays from the 1 2 C (p,YQ) x 3 j j reaction. Therefore, the E2 cross sections could not be determined solely from the present a n a l y s i s ) . The s o l i d and dotted curves are r e s u l t s from a DSD c a l c u l a t i o n and w i l l be discussed further i n the next chapter. I t i s seen that both the second solutions, where they are acceptable, and the solutions obtained with A2 and B1 excluded from the analysis l i e at higher values of a„„ and have larger errors than the solutions found at the "low" (solution I) values of aE2. The "high" values (the second solutions) of a£2 from the analysis with A 2 and Bl included also fluctuate more than the "low" values. The l a t t e r r e s u l t possibly indicates that the "low" values are the correct ones, because the observed smooth variation of the a, and b, c o e f f i c i e n t s k k (see Figure III-7) would follow naturally from a smooth variation of aE2, but would be surprising i f aE2 showed a r a p i d l y varying energy dependence. 76 It i s d i f f i c u l t to know what to make of the fact that the aE2 l i e at systematically higher values when A 2 and Bj are excluded than when they are included. Some of the problems associated with the analysis when they are excluded have been discussed i n reference to the r e p r o d u c i b i l i t y of these c o e f f i c i e n t s . I t should also be noted that s a t i s f a c t o r y f i t s are obtained under the assumption that there i s no Ml radiation present. Unfortunately, the only way to ascertain unambiguously i f M1 r a d i a t i o n i s present i s to measure the plane p o l a r i z a t i o n of the outgoing photon produced by a reaction i n i t i a t e d by a polarized proton (BU 75b). Such a measurement i s not currently experimentally possible. 77 Chapter IV THE DSD MODEL, SUM HOLES AND COMPARISON BITH OTHER EXPERIMENTS The d e s i r a b i l i t y of having reaction models available with which to compare experimental data was mentioned i n Chapter I. This i s e s p e c i a l l y true in (p,Y) reaction studies of E2 strength where, even with polarized beams, there i s not usually enough information available to separate the E1 and E2 components. Moreover, proton radiative capture reactions are p a r t i c u l a r l y s e n s i t i v e to the presence of d i r e c t E2 components (HA 73b), so in cases where i t i s desirable to learn about the c o l l e c t i v e E2 components (for example, to better understand e f f e c t i v e charges) i t i s necessary to have a model available with which comparisons of guantities extracted from the data can be made, since the experiment i t s e l f cannot distinguish between these two components. Thus the purpose of a reaction model i n the p a r t i c u l a r case of (p,Y) reactions i s to predict what e f f e c t s the presence of d i r e c t and/or c o l l e c t i v e E2 strength w i l l have on the experimentally measured guantities. Then a comparison of the predictions with the actual experimental observations w i l l give information on the extent to which the assumptions about the E2 strength are j u s t i f i e d . Following the work of Potokar and others (PO 73 and references therein), Snover and Ebisawa (SN 75) have recently extended the dir e c t semi-direct capture model to include d i r e c t and 78 c o l l e c t i v e E2. The model w i l l fee described b r i e f l y , and comparisons of the model predictions to the present data w i l l be made. .. This chapter also contains comparisons of the present r e s u l t s to the is o s c a l a r EWSR, and to the results of other, si m i l a r experiments. 4.1 The DSD Model Lane and Lynn (LA 59) considered a direct capture model to explain the observation of Cohen (CO 55) that the cross sections for (p,Y) reactions i n the energy range from 8 MeV to 22 MeV were approximately constant. These cross sections were expected to be f a l l i n g rapidly as a function of energy i f the compound nucleus model of Bohr (BO 36) was v a l i d . Lane and Lynn considered the case where the incoming proton radiated energy and was captured d i r e c t l y into a bound state before a compound nucleus was formed. The process was considered to be mainly extra-nuclear, so that the d e t a i l s of the nuclear i n t e r i o r were r e l a t i v e l y unimportant. Although t h i s model gave cross sections that were i n order of magnitude agreement with experiment, they were s t i l l too small by a factor of about four. Brown (BR 64) extended t h i s model to include the case where the incoming proton excited the target nucleus into i t s giant (dipole) resonance state and the proton was then scattered into a bound state. The excited core of the nucleus 79 plus bound proton system then de-excited from the c o l l e c t i v e GDB state by emitting a gamma ray. Brown termed t h i s process Msemi-direct" to distinguish i t from the one step direct process considered by Lane and Lynn. The two processes are pictured schematically in Figure IV-1. Other workers have considered the effects of a semi-direct amplitude i n nucleon capture reactions. Clement, Lane and Book {CL 65), i n a treatment only s l i g h t l y d i f f e r e n t from Brown»s, showed that the inclusion of the semi-direct process improved the agreement between the calculated and measured cross section for the reaction 1* 2Ce (p, Y) 1 v 3 P r from 10 MeV to 50 MeV. Longo and Saporetti (LO 68) included the interference term between the d i r e c t and semi-direct parts and found further improvement i n the calculated cross section for t h i s experiment. In a subsequent paper, Longo and Saporetti (LO 69) showed that inclusion of a d i r e c t E2 amplitude was important for energies above 20 MeV. More recent studies with the model have investigated d i f f e r e n t approaches to handling the description of the c o l l e c t i v e excitation (ZI 70, PO 73)., Snover and Ebisawa have extended the DSD model to include direct and c o l l e c t i v e E2 amplitudes. Because of the new d e t a i l s being measured i n (p,Y) reactions, they calculate, i n addition to the cross section, the angular d i s t r i b u t i o n s of the cross section and of the analyzing power. In order for a reaction model to be considered successful, i t should be able to describe s a t i s f a c t o r i l y a l l of these experimentally measurable quantities. An outline of the model follows. The d i f f e r e n t i a l cross section for a process undergoing a 80 ENERGY initial P + A COLLECTIVE A+ I final Fig. IV-1 : Schematic representation of d i r e c t and semi-direct processes. The i n i t i a l p + A scattering system can proceed d i r e c t l y to the f i n a l bound A + 1 state with the emission of a gamma ray, or i t can f i r s t excite the GDR of the core before the proton i s captured into a bound state. The core • bound proton system then de-excites by the emission of a gamma ray. 81 t r a n s i t i o n from an i n i t i a l state i to a f i n a l state f i s given by (ME 65b) IV-1 In t h i s expression, <f>i and <t>f are the i n i t i a l and f i n a l state wave functions, respectively, v i s the incident p a r t i c l e v e l o c i t y , p(E) i s the density of f i n a l states and £ i s the appropriate e l e c t r i c multipole operator. Thus ca l c u l a t i o n of the d i f f e r e n t i a l cross section reduces to calcul a t i n g matrix elements of the form M.^f = «frf | £!<!,.> IV-2 Now the Hamiltonian for the interaction of the incident nucleon with the target nucleus can be written as (BS 59b) H = H? + T(r) + V(r , t ) IV-3 where H i s the Hamiltonian for the A nuclear p a r t i c l e s T(r) i s the k i n e t i c energy of the incident p r o j e c t i l e and V ( r , t ) i s the sum of the inte r a c t i o n potentials between the incident nucleon at location r and each of the target p a r t i c l e s , whose location i n t o t a l i t y i s represented by 5. A solution to the Schroedinger eguation i s given by $(r,5), where H*(? , t ) = E*(r , t ) IV-4 and the wave function V s a t i s f i e s 82 H f(r\t) = m(r,t) IV-5 o where H = H + T (r). o £ The potential V i s too complicated to permit an exact solution to IV-4, so the complex o p t i c a l model potential V ^ i s introduced- where opt V = V + 6V IV-6 opt and 6V, the residual particle-hole i n t e r a c t i o n , i s treated as a perturbation. It was consideration of the interaction 6V that enabled Brown and B o s t e r l i (BR 59a) to co r r e c t l y calculate the energy of the GDR i n the s h e l l model (see Chapter I ) . Adding the o p t i c a l potential to the free p a r t i c l e Hamiltcnian EQ enables a distorted wave function ¥ o p t ( r , f ) to be found from the solution of H° P V p t(r \ f ) = ET°pt(?,l) IV-7 where H o p t = H + V o opt I t i s shown in Messiah (ME 65b) that the solution to IV-4 s a t i s f i e s *(?'^ = (l + l i i T ' 6V) * ° P t < ^ ) IV-8 where e + 0 i n the l i m i t . Substitution of IV-8 into IV-2 gives 83 M. . = <^|£|T°P t>.+ V f ' •' 1 IV-9 l+f f 1 1 l i-, E - H ± i e A where the |<|>^> are the intermediate c o l l e c t i v e states. Now the e l e c t r i c multipole operator e can he s p l i t into two parts S - * N + V i v - t o where e N acts on the nucleon and e T acts on the target., In addition, the |<j) > have, in the present case, only one well defined state |(|>R> of energy and width r R for a given multipolarity t r a n s i t i o n . Therefore, IV-9 f i n a l l y becomes , , O D t ^ f ' ^ ' V «!'Rl6Vl4'?Pt> M = <w e U P > + - 1: • - - IV-11 W i + f - < * f ' e N r i + E - E + . i r / 2 The f i r s t term i n IV-11 represents the di r e c t capture process, and the second term the semi-direct one. Following the d i r e c t capture ca l c u l a t i o n s of Donnelly (DO 67), Snover and Ebisawa expanded the i n i t i a l state wave functions i n terms of r a d i a l , angular momentum and spin wave functions as usual. Since the i n i t i a l state consists of a target i n i t s ground state and an incoming nucleon, and the electromagnetic operator consists of a sum of one-body operators, Snover and Ebisawa carry out a f r a c t i o n a l parentage expansion of the f i n a l bound state at the beginning. This selects out only those parts of the f i n a l state which have parentage in the i n i t i a l state, and therefore s i m p l i f i e s the angular momentum algebra. Upon reduction of the resultant 84 angular momentum algebra, they obtain an expression for the d i f f e r e n t i a l cross section of the di r e c t capture part which i s written i n terms of the reaction matrix elements (E t of section 3.6) and a Legendre polynomial expansion in the angle of the emitted gamma ray. The reaction matrix elements are proportional to a direc t r a d i a l matrix element, fif., via 13 various s t a t i s t i c a l and phase space factors and angular momentum coupling c o e f f i c i e n t s , where i s given by / ,X |,(r) U (r) R i J ^ } = J f = f ( r ) I V " 1 2 where X^j( r) a n ^ \j^r^ a r e t l i e r a d i a l parts of the i n i t i a l and f i n a l state wave functions, respectively, and f^ (r) i s the appropriate d i r e c t e l e c t r i c multipole operator. The ^ j ( r ) are normalized by a phase factor e±a^ , where i s the Coulomb phase, and ULJ{r) i s normalized by the spectroscopic factor for the f i n a l state. With the inclu s i o n of the semi-direct part of IV-11, the r a d i a l matrix element i s modified to U L J ( r ) A / r F f T ( r ) T* R 2 D R where a ^ T i s the strength with which the given resonance of order <£, isospin T, i s excited and ^-T^r^ ^ s a f ° r n 3 factor which describes the manner i n which the c o l l e c t i v e state i s excited. Snover and Ebisawa (SN 76) choose a hydrodynamic model 85 form factor for ^(r)» that i s , the c o l l e c t i v e model description of the GDR i s taken to be that of an o s c i l l a t i o n of a l l the protons in the nucleus against a l l the neutrons in the nucleus, with the nuclear surface remaining r i g i d . In t h i s case, - F - Q C * ) i s given by rV^r) where V^rJ/4 i s the r e a l symmetry term i n the o p t i c a l p o t ential. For proton radiative capture. a i s given by 3h|ze n 2VP IV-14 "11 4M A<rz>E_. P 11 where & i s the f r a c t i o n of the c l a s s i c a l dipole sum rule (equation 1-1) exhausted by the resonance of energy E l l and <r 2> i s the mean squared radius of the charge d i s t r i b u t i o n in the nucleus. The extension of the model to include dir e c t E2 i s straightforward and involves, for example, using X - 2 f o r the direct form factor ( r ) . C o l l e c t i v e isoscalar E2 strength i s introduced by using a form fa c t o r F^r^r) given by dV (r) F = -r — IV-15 20 dr where VQ{r) i s the r e a l central nuclear potential. The strength c v f T i s given by * 2 p 2 0 a20 =2M~E7 n I V ' 1 6 p 20 where 32Q i s the f r a c t i o n of the EHSR (eguation 1-2) exhausted by the E2 resonance. The guantities E^ T» a n <* r^ T a r e n o t s p e c i f i e d by the model. They are adjusted to f i t the t o t a l cross section as 86 well as possible. 4.2 The Optical Model and Calculation of the Wave Functions The r a d i a l parts of the i n i t i a l scattering state wave functions, x„.(r), are calculated in the o p t i c a l model, f i r s t S.3 proposed by Fernbach, Serber and Taylor (FE 49) to describe the scattering of incident p r o j e c t i l e s off a nucleus. The o p t i c a l potential i s given by V = v + V + V TV-17 opt , CN SO coul A V ' ' where V C N and V S Q are the (complex) central nuclear and spin o r b i t potentials, respectively, and V c o u l i s normally taken to be that of a uniformly charged sphere of radius B c, and i s given by —I 3-**) F O R R < R c \ ' v i coul Z Z e 2 1 T — for r > R c where z and Z T are the atomic numbers of the incident and target n u c l e i , respectively. Normally to ca l c u l a t e the wave functions, o p t i c a l potentials taken from e l a s t i c scattering data involving the incident nucleon (in t h i s case, a proton) and the target 87 nucleus under consideration ( l 2C) would be used. Unfortunately, no o p t i c a l potential exists which s a t i s f a c t o r i l y describes both the d i f f e r e n t i a l cross section and analyzing power of the reaction 1 2 C { p , p o ) i 2 C in the energy region of interest here. . The problem i s that both of these guantities vary rapidly as a function of energy; thus the smoothly varying o p t i c a l potential i s not able to match the f l u c t u a t i o n s . The rapid variations are caused by the dominance of resonance structures throughout t h i s region, as shown i n the data of Meyer et a l . (ME 76) . ftn attempt to describe the e l a s t i c scattering of protons from carbon from E-> = 12 MeV to 20 HeV was made by Nodvik, Duke P and Melkanoff (NO 62). These authors assumed the o p t i c a l potential to be of the following form; Re(V C N) = ,- V f ( r ) IV-18 Im(V C N) = - W exp{-(r-R) 2/b 2} IV-19 / ^ \ VS df (r) + « vso " - ^ T c j - t e 0 ^ IV"20 f ( r ) = [l + exp{(r-r o)/a}J _ 1 IV-21 where V, W and V s are the depths of the various potential wells and 1i/m c = i/2.0 fm. ir Nodvik et a l . were able to f i t the d i f f e r e n t i a l cross sections and analyzing powers guite well i n d i v i d u a l l y by 88 l e t t i n g several of the parameters i n eguations IV-18 to IV-21 vary, but they could not f i t both guantities well simultaneously. In addition, the parameters fluctuated wildly as a function of energy, thereby v i o l a t i n g the s p i r i t of the o p t i c a l model. Their f i n a l r e s u l t s included compromise potentials which generally did reasonably enough fo r the d i f f e r e n t i a l cross sections, but did not do so well for the analyzing powers. another set of o p t i c a l model parameters was obtained by Watson, Singh and Segel (W& 69). These authors f i t t e d the d i f f e r e n t i a l cross section and polarization data for a variety of p-sheli n u c l e i , including 1 2C, over the energy range from E^ - = 10 MeV to 50 MeV. Their form of the o p t i c a l potential was the same as that used by Nodvik et a l . , except that the Gaussian shape for the imaginary part of the c e n t r a l potential was replaced by a surface derivative form given by Im(V„XT) = 4a TW d f ( r ) IV-22 CN' I dr The diffuseness a^ . of the imaginary part was given a value s l i g h t l y d i fferent from the diffusenesses of the r e a l and spin - o r b i t parts of the potential. The f i t s obtained by Watson et a l . to the data on e l a s t i c scattering from 1 2 C were not, in general, as good as those obtained by Nodvik et a l . This was to be expected (PE 70) , since Watson et a l . were f i t t i n g a wider range of nuclei and their parameters were constrained to be smooth functions of energy, while those of Nodvik et a l . were not. Thus, i t was decided to attempt to find a better set of 89 o p t i c a l model parameters that would describe more s a t i s f a c t o r i l y both the d i f f e r e n t i a l cross section and analyzing power for the 1 2C(p,p ) 1 2 C reaction. The data of o Meyer et a l . (ME 76) were used in the analysis. The s t a r t i n g parameters were taken to be those of Sene et a l . (SE 70), who f i t t e d polarized neutron scattering data from l 2 c at E-* = 14.1 n MeV reasonably successfully. The form of the o p t i c a l potential was taken to be the same as that used by Watson et a l . , except that a l l three of the shapes f (r) (eguation IV-21) were allowed to have independent r a d i i and diffusenesses. The o p t i c a l model code ABACUS-21 was used to perform the calculations. A s l i g h t modification to the code was made so that the shape parameters for the spin-orbit potential could be given values independent of those used for the shape of the central nuclear po t e n t i a l . . By systematically varying the parameters, i t was found that a f a i r l y good f i t could be obtained to the d i f f e r e n t i a l cross section and analyzing power data simultaneously. The parameters which best f i t the data are l i s t e d i n Table IV-1. Also l i s t e d i n t h i s table are the parameters obtained by Watson et a l . , to be referred to as WSS, and by Becchetti and Greenlees (BE 69b), to be referred to as BG. This l a s t set was obtained for a wide range of n u c l e i with A>40, E<50 MeV. Some comments on the new set of parameters follow. Written by E. H. Auerbach at Brookhaven National Laboratory and adapted for use at the University of B r i t i s h Columbia by T. W. Donnelly and A. L. Fowler. 90 Parameter* Table IV-1 Optical Model Parameters Potential- / ( R o ) . ( R o ) SO * R 'SO W ( WSS BG NEW 1.15-.001E 1.17 1.13 1..15-.001E 1.32 1.40 1.15-.001E 1.01 .88 .57 .75 .55 .50 .51 .12 .57 .75 .24 . 60-.28E+.4Z/A1/3 54- .32E+.4Z/A1/3 58.4 .59E (E<15 MeV) .22E-2 .7 (>0) Gaussian 6-.055E (E>15 MeV) 11.8-2 25 •5E (>0) Surface derivative .42-. 5.5 6.2 7 .28-. SO * r a d i i and diffusenesses are in fm, potentials are in MeV, E is the laboratory energy. RQ refers to the coefficient of in r=R0A^3. Terms proportional to (N-Z) have been suppressed since N=Z for 1 2C. 91 The well depths for the new potential (NEW) are f a i r l y reasonable when compared to other o p t i c a l model analyses. The depth of the r e a l central potential i s expected to have an energy dependence because of i t s non-locality and a Coulomb term proportional to ZA - 1/ 3 (PE 63) . These are seen e x p l i c i t l y i n the WSS and BG potentials. However, in the present analysis V fluctuated considerably and an o v e r a l l trend was d i f f i c u l t to detect, so i t was l e f t constant at the value i t had at E^ > = 16.964 8eV. Neither the d i f f e r e n t i a l cross section nor the analyzing power varied too rapidly for a few hundred k i l o v o l t s either above or below th i s energy, so the e f f e c t of resonances i s apparently not too strong. The radius parameters for the imaginary and spin-orbit parts of a l l three potentials are guite d i f f e r e n t . However, i t was found that the extreme values obtained for NEW gave the best f i t s to both the d i f f e r e n t i a l cross section and analyzing power. An increase of 10% i n ( R o ) S 0 t o oring i t more into l i n e with the other analyses worsened the x 2 of the f i t (defined as the sum going over both the d i f f e r e n t i a l cross section and analyzing power data) by about a factor of 20. A decrease in While the value for the diffuseness of the r e a l potential well a R i s s i m i l a r to the other r e s u l t s , a-^  and a g 0 d i f f e r markedly. Actually, the value of &1 i n the present analysis gives a shape to the imaginary part of the potential which i s very s i m i l a r to that obtained by Nodvik et a l . The f i t s P N { R q ) i of 10% worsened x 2 by about a factor of 3. 92 deteriorated rapidly when either a or a was varied from i t s optimum value. Shown in Figure IV-2 are the f i t s to the d i f f e r e n t i a l cross section and analyzing power data at E^ = 16.964 MeV obtained with the Nodvik et a l . (NDM), WSS and NEW parameter sets. The analyzing power data is c l e a r l y f i t much better with NEW, esp e c i a l l y at the forward angles, and t h i s was c h a r a c t e r i s t i c of the f i t s at a l l energies. There i s not much to choose between NDM and NEW for the f i t s to the d i f f e r e n t i a l cross section at t h i s or any other energy; WSS consistently underestimates th i s guantity at a l l the energies. One disappointing aspect of a l l three of these o p t i c a l model sets i s that none of them reproduce s a t i s f a c t o r i l y the p a r t i a l reaction cross sections. Shown in Figure IV-3 i s a compendium of reaction cross section data (ME 77) for the d"3/2 and S j ^ p a r t i a l waves, and the reaction cross sections for these p a r t i a l waves calculated with the various potentials. The potential sets give d 3 ^ 2 reaction cross sections which do turn over s i m i l a r l y to the data, but at a lower energy. The calculated s 1 / 2 p a r t i a l cross sections continue to r i s e at lower energies, contrary to the data. I t i s possible to include the t o t a l reaction cross section i n the x 2 search for the best f i t , but i t i s d i f f i c u l t to know how to weight t h i s guantity (PE 70), and in improving the f i t to the reaction cross section, i t may happen that the guality of the f i t s to the d i f f e r e n t i a l cross section and analyzing power w i l l deteriorate badly. Hence, i t was decided not to include the reaction cross section in the f i t t i n g procedure. 93 •6jm (degrees) F i g . IV-2 : l 2 C ( p , p D ) * 2 C d i f f e r e n t i a l cross section and analyzing power at EJ = 16.964 MeV. The data are from reference (ME 76). The curves are from o p t i c a l model f i t s using the potentials shown. F i g . IV-3 : Comparison of the o p t i c a l model analyses with some experimental p a r t i a l reaction cross sections. The s o l i d curve i s a compendium of data (ME 77). 95 It i s indeed not too surprising that the observed guantities - the d i f f e r e n t i a l cross section, the analyzing power and the p a r t i a l reaction cross section - are not a l l well reproduced by t h i s simple use of the o p t i c a l model. It has been shown i n the work of Mikoshiba, Terasawa and T a n i f u j i (MI 71) that i t i s important to consider the e f f e c t s of coupling between scattering i n the e l a s t i c channel and scattering i n the i n e l a s t i c channel to the 2 + state in *2C at E x - 4.43 Mey. These authors investigated the region from E->- = 4 MeV to 8 MeV with a coupled channel c a l c u l a t i o n and P found that i t was possible to f i t reasonably well the observed exci t a t i o n functions and angular d i s t r i b u t i o n s of both the cross section and analyzing power at these low energies. In order to extend t h i s type of c a l c u l a t i o n to the energies of interest i n the present work i t would be necessary to consider the coupling of other i n e l a s t i c channels i n addition to the one at 4.43 MeV to the e l a s t i c channel. This was shown i n the work of Johnson (JO 74) who extended the calculations of Mikoshiba et a l . to higher energies and found that the model began to break down seriously at about E-> = 10 MeV. A major P computational e f f o r t would therefore be reguired to extend t h i s approach to higher energies and t h i s was not attempted in the present work. A l l three parameter sets (NDM, »SS and NEI) were used to calculate the i n i t i a l scattering state wave functions Xg..(r) and the f i n a l bound state wave function, 0 L J ( r ) . The l a t t e r was found by setting H = 0, and varying V to f i t the binding 96 energy (E B) of a {Pjy2 ^ proton i n the ground state of (E = -1.94 HeV). The r e a l parts of the r a d i a l wave function B calculated for the bound state, the i n i t i a l scattering states S2_/2 a n f l ^3/2 a n < 3 t b e f o r m f a c t o r rV^r) used f o r the c o l l e c t i v e part of the E1 capture, are shown in Figure IV-4. I t can be seen from these plots why the 0 * 3 / 2 wave i s expected to be dominant i n the capture; i t i s because the H a v e n a s a node inside the nuclear i n t e r i o r (the nuclear radius i s shown as *r o') leading to a p a r t i a l cancellation of i t s contribution to the cross section. 4.3 Calculation of the Direct Semi-Direct Capture The f i r s t parameter from a d i r e c t semi-direct capture c a l c u l a t i o n that must be matched to experiment i s the t o t a l cross section. This i s true because most of the t o t a l cross section arises from e l e c t r i c dipole capture, and i f the E1 strength cannot be accounted for c o r r e c t l y , then cal c u l a t i o n s including E2 strength would be of dubious value. Thus the c a l c u l a t i o n of the t o t a l cross sections including the GDR and di r e c t E1 and E2 capture only are shown in Figure IV-5 as a s o l i d dotted l i n e (NDM) , an open dotted l i n e (HSS) and a dot-dash l i n e (NEW). The cross section calculated with the three potentials are indistinguishable i n the region of the GDR near E = 20.8 MeV. The s o l i d curve i s the t o t a l cross section x taken from reference (BE 76b). The position E and width r n were adjusted to match the shape of the GDR, 97 Fig. : The r e a l parts of the r a d i a l wave functions calculated with the NEW p o t e n t i a l . Scattering wave functions <s >2 and d 3 ,2 ) leading to E1 capture only are shown. The form factor r f ( r ) for volume coupling to the GDR i s also shown. r indicates the nuclear radius. ° 98 40 30 20 «3-2 O O UJ to CO CO o ce o o 10 x4 1 1 1 GDR PLUS DIRECT CAPTURE T ONLY : PYGMY INCLUDED 1 1 [—j • NDM POTENTIAL ooooooWSS POTENTIAL -NEW POTENTIAL w s s POTENTIAL J I I L 30 Et Q b (MeV ) F i g . IV - 5 : F i t s to the » 2 C { p , Y 0 ) i 3 N t o t a l cross section. The s o l i d curve i s a f i t (by eye) to the data of Berghofer et a l . (BE 76b). 99 and 3,, was adjusted to match the t o t a l cross section. The 11 symmetry potential V^O) was taken to be 100 MeV from an estimate given by Bohr and Mottelson (BO 69c), and the mean square radius <r2> was taken to be 6.0 fm 2 from electron scattering r e s u l t s on l*N (ME 59) (<r2> i s approximately constant for p-shell nuclei (PH 75)) . It i s clear that a l l three p o t e n t i a l sets give no indication of the presence of the pygmy resonance. Presumably, t h i s i s because the coupling with the i n e l a s t i c channels i n the calculation of the wave functions has been ignored. Note also that the pygmy resonance shape somewhat resembles the shape of the ^3/2 p a r t i a l absorption cross section (Figure IV-3). Failure to reproduce the l a t t e r might have some ef f e c t on the f a i l u r e to reproduce the pygmy i f the pygmy i s mainly a d 3^ 2 resonance. Some evidence supporting the importance of considering the coupling of i n e l a s t i c channels in calculations of the (P, Y 0) cross sections i s found i n the work of Johnson (JO 74), who did a coupled channel c a l c u l a t i o n for the l 2 G ( p , Y 0 ) 1 3 N reaction for energies up to E p = 9 MeV. This i s below the lowest energy measured in the present work, but i t does extend i n t o the low energy t a i l of the pygmy resonance. His c a l c u l a t i o n of the 90° y i e l d curve, which r e s u l t s mostly from E1 capture, reproduces the experimental measurements very well. However, i t has already been stated that his model begins to break down near the energy where the present measurements begin, so more complicated couplings would have to be introduced. Therefore, i n order to reproduce the presence of the pygmy 100 resonance, i t was necessary to introduce a second c o l l e c t i v e E1 amplitude into the present c a l c u l a t i o n . This has sometimes been necessary i n previous c a l c u l a t i o n s with the DSD model; for example, Snover et a l . {SN 76) provide for fragmentation of the GDR in i SN by including a small second resonance amplitude. The dashed l i n e i n Figure IV-5 shows the result of including a second amplitude i n the present case using the WSS p o t e n t i a l . The r e s u l t s of the c a l c u l a t i o n for a l l three potentials agreed to within ±2055 for the t o t a l cross section and also for most of the other quantities calculated by the model, so only the r e s u l t s using WSS (the most general of the three potentials) w i l l be referred t o i n future. The only exception was that HEW tended to give E2 direct capture cross sections that were about U0% larger than NDM or WSS above E+ = 16 HeV. The parameters used to reproduce the t o t a l cross section shown i n Figure IV-5 are l i s t e d i n Table IV-2. Table IV-2 GDR Parameters Used to Reproduce the Total Cross Section Using the WSS Potential E l l r i l P l l GDR 20.5 MeV 4.0 MeV 0.6 Pygmy 13.8 MeV 6.0 MeV 1.8 The very large value f o r g required to reproduce the pygmy resonance indicates that the pygmy i s not a fragment of the GDR. I t i s not the r e s u l t of isospin s p l i t t i n g of the GDR, for example. Nevertheless, reference to the figures shown i n 101 Chapter III indicates that the model i s reasonably successful in reproducing most of the experimental quantities. The s o l i d l i n e s in these figures represent the c a l c u l a t i o n referred to above. In cases where a dotted l i n e i s shown, an isoscalar E2 resonance i s assumed to l i e at E •= 25 MeV {see below). X In Figure III-7, i t can be seen that the calculations follow the trend of the measured angular d i s t r i b u t i o n c o e f f i c i e n t s f a i r l y well. The magnitude of the calculated a 3 i s possibly a l i t t l e low, and b^ i s perhaps more constant than the c a l c u l a t i o n indicates. The c a l c u l a t i o n also f a i l s to reproduce the broad structure i n a 2 ; t h i s i s very l i k e l y related to i t s f a i l u r e to reproduce the s,d phase difference as seen in Figure III-8. The d-wave and s-wave amplitudes are well reproduced for the s o l u t i o n I values however, and the calculated phase difference represents a f a i r l y good average value of the measured one. Nothing i s present i n the model as i t stands t o produce such a phase v a r i a t i o n ; here again, ca l c u l a t i n g the wave functions with allowance for coupling to the i n e l a s t i c channels would be very i n t e r e s t i n g . Neither the magnitude of the p/f r a t i o nor the p,f phase difference i s given c o r r e c t l y by the model although the differences are not too severe. This i s seen i n Figure III-9, where i t i s also noted that the magnitude of the p,d phase difference i s a l i t t l e low. The f,d phase difference i s well reproduced beyond the structure that i s observed between 10 MeV and U MeV. F i n a l l y , i t can be seen i n Figure 111-12 that direct E2 capture alone s a t i s f a c t o r i l y accounts for the experimentally 102 measured cross sections i f the "low" consistent set of sclutions are the correct ones. The dotted l i n e s in the figures are the r e s u l t of assuming that an i s o s c a l a r E2 resonance e x i s t s at the expected energy of E^ = 25 Me? {see Chapter I ) . In the c a l c u l a t i o n , the width was taken to be the same as that of the GDR ( r 2 o = 4 M e ¥ ) # and the E2 resonance was assumed to exhaust 505? of the sum rule { B 2 Q = 0.5). The central nuclear potential was taken to be VQ(0) = -50 HeV {BO 6 9c). Over most of the region. Figure III-12 shows that the cross section i s r e l a t i v e l y i n s e n s i t i v e to the presence of such a resonance, but at E->- = 16 HeV and 17 HeV, P the c a l c u l a t i o n i s i n disagreement with the data. , It i s shown i n Figure III-7 that the assumption of an E2 resonance brings the calculated a^ into better agreement with the data, but the calculated a^ becomes larger than the present measurements., Overall, there appears to be no need to incorporate c o l l e c t i v e E2 strength into the c a l c u l a t i o n , since the inclusion of only direct E2 amplitudes provides a reasonable description of the angular d i s t r i b u t i o n c o e f f i c i e n t s and the extracted E2 cross sections. 4.4 Sum Rules Another test for the presence of c o l l e c t i v e strength i n a reaction i s that an appreciable f r a c t i o n of the appropriate sum rule should be exhausted {BE 76a). Of course i t i s also necessary that t h i s strength be concentrated in a s u f f i c i e n t l y 103 narrow energy range that i t w i l l appear as a resonance, which does not seem to be the case for the present measurements. Nevertheless, the d i r e c t capture contribution to the sum rules can often be s i g n i f i c a n t . , After converting the measured (p,^) E2 cross sections to the inverse ( Y,P o) E2 cross sections using the detailed balance theorem {Appendix A) the i n t e g r a l of eguation 1-2 from E+ = 10.0 MeV to 17.0 MeV {E =11.1 MeV to p X 17.6 MeV) was found to be 0.44 ± 0.17 yb/MeV, corresponding to 10.3 ± 4.0% of the EHSB {Appendix A). These errors include the ±20% o v e r a l l normalization error i n the determination of the t o t a l cross section by Berghofer et a L , {BE 76b). The f r a c t i o n of the EwSB exhausted by the calculated d i r e c t capture E2 cross section (HSS potential) i s 6.8%. Thus the f r a c t i o n of the ESSB exhausted i n the 1 2 C { p , Y Q ) i 3 N reaction i s consistent with the calculated f r a c t i o n assuming the E2 part of the reaction i s proceeding s o l e l y by dire c t capture. 4.5 Comparison with Other Work The E2 strength that i s extracted from the present measurements i s very t y p i c a l of that found in other {p,^) reactions. Detailed comparisons can be made with the results of studies of the 1 4 C ( p , Y0)»s$j a n c [ i s N ( p , Y 0 ) i * o reactions, since complete analyses of these reactions are available (SN 76, AD 77). In the 1 4 C < p , Y Q ) * r e a c t i o n , the t o t a l E2 cross sections are t y p i c a l l y 1.0 yb from E = 19.5 MeV to 27.0 MeV, compared to E2 cross sections of the order of 0.2 yb from 104 E x = 11.1 MeV to 17.6 MeV i n the present study. However, because of the energy-sguared factor i n the denominator of the EWSR, both reactions exhaust s i m i l a r amounts of the sura rule l i m i t ; (10.3 ± 4.0)% of the sum rule l i m i t i s depleted i n the present case compared to (6.8 ± 1.4)% i n the case of i * C ( p , Y 0 ) i 5 N . A somewhat d i f f e r e n t s i t u a t i o n possibly exists in the (Y,Po) channel of the photodisintegration of l 6 0 . In a study of the i s N ( p , Y o ) i 6 0 reaction, Hanna et a l . (HA 74a) found evidence for a GQR which exhausted approximately 30% of the EWSR between E x = 20.2 MeV and 26.8 MeV (about 7% of the EWSR i s exhausted for the calculated E2 direct capture through t h i s region (SN 76)). Because of the large difference i n the concentration of E2 strength found in th i s reaction from that found in the 1 * C ( p , Y 0 ) 1 S N reaction, Adelberger et a l . (AD 77) remeasured the l s N ( p , Y 0 ) 1 6 ° reaction from E+ = 8 MeV to 18 MeV (E x = 19.6 MeV to 29 MeV). Considerably l e s s E2 strength was found i n the new measurement, although there was s t i l l an excess over d i r e c t capture near E x = 20.6 HeV and 24.8 MeV. It should be pointed out that i n the analysis of the o r i g i n a l data, Hanna et a l . excluded A 1 and Bj from the f i t s to the T-matrix elements and so the E2 cross sections obtained were presumably subject to the li m i t a t i o n s discussed at the end of section 3.8. A l l three of these (p, Y 0) reactions show a great many s i m i l a r i t i e s . , The a^ and b^ c o e f f i c i e n t s show roughly the same trends; a^ . and b^ both become slowly more positive with increasing excitation energy, for example, and a Q becomes more 105 negative. As a r e s u l t of the s i m i l a r i t i e s in the angular d i s t r i b u t i o n s , the behaviour of the T-matrix elements i s much the same. M l three reactions are a f f l i c t e d with the presence of secondary solutions some of which are acceptable i n a s t a t i s t i c a l sense. In f a c t , in the case of the 1 *C (p, YQ) I S J J reaction, some of the second solutions were "preferred" i n the sense that they had lower chi-sguares than those for solution I. A l l three reactions show the same behaviour when and B 1 are excluded from the analysis; namely, the extracted E2 cross sections are systematically higher and the A-^  and B ^  c o e f f i c i e n t s that are reproduced from the analysis which excluded them are consistent with the measured values only when the errors of the reproduced c o e f f i c i e n t s are taken, in t o account. F i n a l l y , i f the solution I values for a £ 2 are taken to be correct, then a l l three excitation functions are s a t i s f a c t o r i l y reproduced by considering only d i r e c t E2 capture <with no c o l l e c t i v e E2 capture) i n t e r f e r i n g with d i r e c t and c o l l e c t i v e E1 capture. This re s u l t i s i n accord with (p,^ 0) measurements i n other l i g h t n u c l e i . For example, Noe et a l . (NO 76) find by comparison to calculations with the DSD model, that there i s no evidence for c o l l e c t i v e E2 strength above the GDH in the l l B ( p , Y 0 ) i 2 c reaction. The amount of E2 strength seen in the present study i s also similar to the strength seen i n many (a,Y) reactions. Most of the information about these reactions comes from ( a , Y Q ) studies on nuclei with ground state J = 0 , since only natural parity states can then be formed i n the compound system, 106 thereby permitting an unambiguous determination of the E1 and E2 strengths through measurements of the angular d i s t r i b u t i o n s . For example, Snover et a l . (SN 74) studied the reaction 1ZC{a,Y0) i&0 and found 17% of the EWSR was exhausted between E =12 MeV and 28 MeV. , Similar r e s u l t s are obtained for other x ( a , Y 0 ) studies on spin 0 nuclei ranging up to A = 60; namely, about 1%/MeV of the EWSR i s exhausted (HA 74c). Thus, the present measurement of the E2 strength i n 1 3 N i s i n agreement with s i m i l a r capture reaction measurements i n other l i g h t n uclei. I t i s also i n agreement with a variety of i n e l a s t i c scattering measurements on the near-by nuclei 1 2 c and l 6 0 . As mentioned i n the introduction, the i n e l a s t i c scattering studies on these nuclei have shown the guadrupole strength to be very much spread out. I t should be noted, of course, that the present experiment investigated only the < Y»P 0) channel i n the decay of 1 3N. Many other possible decay branches e x i s t , f or example, proton decays to excited states in i 2 C and neutron, alpha and deuteron, etc. decays to various le v e l s in other neighbouring nuclei, although many of these channels are not open u n t i l higher excitation energies are reached. Moreover, there often appears to be l i t t l e strength i n the other charged p a r t i c l e channels., For example, Weller and Blue (WE 73) investigated r a d i a t i v e deuteron capture by ll3 from E = 19.5 MeV to 22.3 MeV. They found that the t o t a l (Y,d ) x ' • o cross section was 13% of the t o t a l ( Y,P o) cross section measured i n the i*C(p,Y 0)* 3N reaction (FI 63). I f E2 deuteron decay exhibits the same c h a r a c t e r i s t i c s as E2 proton and alpha 107 decay { a_, = 1% a _ , 0 ) , then the amount of E2 strength in the deuteron channel must be very small indeed. Other n u c l e i near mass 13 studied with (d,^ 0) reactions include *5N (DE 76) and J 1 B (DE 74). Some evidence f o r Ml or E2 radiation occurring i n the region of the GDR of these reactions i s evident from non-zero c o e f f i c i e n t s , but no guantitative estimates were made. Si m i l a r l y , small amounts of E2 or Ml strength are seen i n 3He capture reactions on 1 2 C and l*0 (SH 74) through observation of non-zero a^ c o e f f i c i e n t s . Snover and Ebisawa (EB 77) have recently found a contribution of about 0.3% of the EWSR i n the ( 3 H e , Y 0 ) » 6 0 reaction from E x = 24 MeV to 38 MeV. The present r e s u l t s , then, indicate that the E2 strength in l 3N i s very spread out. No other guantitative measurements have been made in the mass 13 nuclei, but Shin et a l . (SH 71) observed small non-zero a 1 and a 3 c o e f f i c i e n t s i n photoproton angular d i s t r i b u t i o n s from i n e l a s t i c electron scattering on l 3C at excitation energies ranging from 21 to 32 MeV. Shin et a l . also observed much larger a^ c o e f f i c i e n t s i n a simi l a r study on l 2 C . These authors point out that the GDR i s more concentrated i n * 2C and thus the effects of E2 interference w i l l be seen more c l e a r l y i n the t a i l regions of the GDR of th i s nucleus than w i l l be seen in the t a i l regions of the more spread out GDR of 1 3C. In e l a s t i c alpha scattering on 1 2 C (KN 76) has shown that less than 20% of the EWSR i s exhausted between E =15 MeV and 30 MeV i n * 2C, and so i t would seem that the present 1 2 C ( p , ^ ) 1 3 N measurement has accounted for a considerable portion of the E2 108 strength expected in mass 13, esp e c i a l l y noting that the present cross sections are actually lower l i m i t s . 109 Chapter V SUMMARY AND CONCLUSIONS The main purposes of the present measurements were to extend our knowledge of the nucleus 1 3 N by investigating the nature of the E2 strength in the region below the GDR and to provide a simple test of the DSD capture model of Snover and Ebisawa. These aims have been largely achieved, but for a variety of reasons to be given below, success has not been complete., I t would appear that capture gamma ray reactions induced by polarized protons do not permit as unambiguous a determination of the capture amplitudes as was previously believed. This had already been shown to be true for the * * C ( p , Y 0 ) i 5 N and l s N ( p , Y Q) * * o reactions, where the measurements were i n the region through and above the GDR. The resu l t s from the 1 2 C ( p , Y 0 ) i 3 N reaction reported here extend these uncertainties to the region below the GDR as well. The problem i s twofold; two solutions exist at several energies which are both s t a t i s t i c a l l y acceptable, and there might be seme Ml radiation underlying the structure i n thi s region. So far as the f i r s t problem i s concerned, there are several reasons to prefer the solution I r e s u l t s . F i r s t , these solutions give an E2 cross section which varies more or less smoothly with energy and the observed smooth variation of the 110 angular d i s t r i b u t i o n c o e f f i c i e n t s would arise naturally from t h i s , but not from a cross section that was fluctuating widely. In addition, the second solutions are not always acceptable i n a s t a t i s t i c a l sense, and those which are not cannot be the physical solutions. F i n a l l y , the second solution cannot always be found, for example at E+ = 12.8 MeV, although t h i s may be the f a u l t of the i n i t i a l search parameters. Thus a soluti o n which comes and goes might be believed to be just a mathematical solution which has no physical s i g n i f i c a n c e . An attempt to learn about the possible presence of M1 strength was made by performing an analysis i n which the c o e f f i c i e n t s most sensitive to M1 radiation, and B^, were excluded. In t h i s analysis, systematically higher values of the E2 cross section were found, although these values also had much larger errors. The reason for the increase in the errors i s clear enough - the £2 amplitudes were being extracted from a data set that was l e s s sensitive to the i r presence. However, i t i s not clear why the amplitudes were always larger. A major drawback to t h i s analysis was that the valuable x 2 s i g n i f i c a n c e test was lost since there were zero degrees of freedom in the f i t . In any event, the solutions determined from using a l l of the c o e f f i c i e n t s lay at s t a t i s t i c a l l y s a t i s f a c t o r y l e v e l s , so again these are the preferred solutions. Accepting the low, consistent set of E2 cross sections as being correct, the re s u l t s of t h i s experiment agree with many measurements made in other l i g h t n u clei. In p a r t i c u l a r , proton r a d i a t i v e capture i n t o * 2C, *3N, l 5N and **0 bear many s i m i l a r i t i e s and exhaust roughly equivalent amounts of the 111 EWSR. The l a t t e r r e s u l t has been observed in a variety of reactions involving l i g h t n u c l e i . Thus the present measurement has further confirmed the systematics of E2 photodisintegration f o r low k nuclei. The c a l c u l a t i o n s with the DSD model suffered from the deficiency that the pygmy resonance did not appear naturally from the ca l c u l a t i o n . This was actually an expected r e s u l t since Johnson's work showed the importance of the coupling between the ground state and the f i r s t excited state of 1ZC i n reproducing the pygmy*s low energy t a i l . I t might be noted that several attempts to describe the pygmy with s h e l l model calculations have not been wholly successful (Kl 74, Jk 71). However, when the presence of the pygmy was a r t i f i c i a l l y introduced into the present c a l c u l a t i o n , the DSD model was reasonably successful in reproducing the parameters related to E2 capture. In p a r t i c u l a r , the energy dependences of the odd a^ and c o e f f i c i e n t s were s a t i s f a c t o r i l y calculated assuming that only E2 direct capture radiation was i n t e r f e r i n g with the dominant E1 radiation. The various phases associated with the E2 amplitudes were also given reasonably well by the model. The E2 d i r e c t capture cross sections calculated with the model agreed with the measured cross sections, assuming the solution I values were the correct ones. In summary, no strong evidence for the existence of a GQR located in the region of the GDR i s found either in the experimental measurements of the E2 cross section or i n comparisons of the re s u l t s with the DSD model. The E2 cross section contributes an amount to the EWSR (10.3 ± 4.0%) that i s 112 si m i l a r to the strength seen i n a variety of <p,Y0) and (a, YD) reactions in the region below the expected GQB. It would be of i n t e r e s t to extend these measurements to higher energies to search for evidence of c o l l e c t i v e E2 strength. Extension of the measurements both above and below the region studied in the present work using a f i n e r grid would be of interest to establish whether the "resonances" and "dips" observed in some of the extracted quantities are r e a l or merely s t a t i s t i c a l . I t would also be of in t e r e s t to extend a coupled channel c a l c u l a t i o n through the pygmy resonance region to try to understand i t s structure in a more sat i s f a c t o r y way. Much detailed experimental information now exists with which comparisons of such a c a l c u l a t i o n could be made. 113 Bibliography AD 73 E. G. Adelberger, M. D. Cooper and H. F. Swanson, University of Washington Annual Report (1973) p.11. „ AD 77 E. G. Adelberger, J. E. B u s s o l e t t i , K. Ebisawa, R. L. Helmer, K. A. Snover, T. A. Trainor, University of Washington Annual Report (1977) p.43. AJ 70 F. Ajzenberg-Selove, Nuclear Physics A, 152 (1970) 1. AR 75 E. D. Arthur, D. M. Drake and I. 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Madsen, Physics Letters B, 33 (1970) 205. 121 Appendix A THE ENERGY WEIGHTED SUM RULE In t h i s appendix, the means by which the EWSR i s evaluated w i l l be presented. This expression was f i r s t developed by Gell-Mann and Telegdi {GE 53) for even-even nuclei. The cross section for an e l e c t r i c t r a n s i t i o n of freguency w and order L i s given by (OC 73) 2L-1 cr(uL) = (27r) 3e 2 •<™>(t) " L{(21+i)i!;! }'z B(EL,u) A-1 where B{EL,U) i s the reduced t r a n s i t i o n probability. Setting E = Itw, eguation A-1 becomes „ 2L-.1 a (EL) = (2iT) 3e 2 <"»(s) ' L{(2L+1)!!} z B(EL,E) which reduces to A-2 a(E2) = ^ ^ 3 e 2 ( ^ ) B(E2,E) A-3 for L = 2. Now use i s made of the well known sum rule concerning the reduced t r a n s i t i o n probability and the energy summed over a l l f i n a l states f. This i s given by (NA 65) £ B(EL,E)] - L(2L+D 2 -h 2Z 2 2L-2 4TT 2MA <r A - a 2L-2 where <r > i s the {2L-2)th moment of the charge d i s t r i b u t i o n in the ground state of the nucleus, M i s the nucleon mass and A i s the atomic number of the nucleus. 122 With L = 2, equation A-4 becomes Thus, using eguation A-3, the EWSR given by A-6 reduces to a'(E2) Tr 2e 2 Z 2 <r 2> Upon setting e 2/hc = 1 / 1 37 and putting Z = A / 2 , the Gell-Mann-Telegdi result i s obtained as o(E2) TT2 A <r 2> d E " 137 12 "M^ A ~ 8 The oft-quoted sum r u l e J g ( E | ) dE = 0. 2 2 Z 2 A " V 3 yb/MeV follows from putting <r2> = 3/5 R 2 , where R = R Q A V 3 and R = 1.2 fm ( i . e . the charge d i s t r i b u t i o n i s assumed to be o uniform throughout the nucleus). To obtain the EWSR for the nucleus 1 3N, the expression given i n eguation A-7 i s used. The mean sguare radius of the charge d i s t r i b u t i o n for l 3 N was assumed to be the same as the value for **N. From the work of Meyer-Berkhout et a l . (ME 5 9 ) , t h i s i s 6.0 fm 2. Thus from equation A - 7 , the EWSR l i m i t i s 4.29 yb/MeV. To evaluate / dE for the experimental measurements, i t i s f i r s t necessary to convert the capture (p, Yo) cross sections to the inverse ( Y*P Q) photodisintegration cross 123 sections. This i s accomplished using the pr i n c i p l e of detailed balance, which states that {DE 67) (21 +1)(21 +1) p 2 a ( Y , P o ) = o (p ,Y ) 2 ( 2 I + 1 ? - JZ A-9 A * l where I T , I and I A are the spins of the target nucleus, the proton and the residual nucleus, respectively, in a (p,Y) experiment and Pj and p 2 are the centre of mass momenta of the incident p a r t i c l e s i n the {*,p) and (p,Y) processes, respectively. Substitution of the quantities appropriate to the 1 2 C ( p , ' 0 ) 1 3 N reaction into equation A-9 yields 793.3 E L a b o(Y,Po) - g-T"2— a(p,YQ) yb/MeV A-10 ° Y ° After the E2 photodisintegration cross sections had been determined in th i s manner, the EWSR was evaluated by breaking up the energy region studied into eight segments. Each of these segments was bounded by energies where experimental measurements were made. . The EWSR was then calculated i n each segment and the r e s u l t s added together to y i e l d 17.6 MeV / dE = 0.44 ± 0.17 yb/MeV A-1 1 11.1 MeV Further discussion of t h i s result i s given in Chapter IV. 124 Appendix B T-MATBIX ELEMENTS In t h i s appendix, the expressions connecting the angular d i s t r i b u t i o n c o e f f i c i e n t s to the reaction amplitudes and phases w i l l be developed. Transitions involving M1 radiation w i l l be considered in addition to those involving E1 and E2 radiation. It has already been stated in Chapter III that for the case of a 0 + target, s ^ and i n c o n , i - n < ? p a r t i a l waves lead to E1 capture, and a n < ^ f5/2 w a v e s lead to E2 capture. These states w i l l be abbreviated as s, d, p and f, respectively., By angular momentum and parity conservation, p 1y 2and P 3 / 2 P a r t i a l waves lead to M1 capture. Making use of the tables of Carr and Baglin (CA 12), the connection between the above reduced matrix elements and the unpolarized angular d i s t r i b u t i o n c o e f f i c i e n t s can be written down immediately. Using eguations III-5 and III-9, i t i s a t r i v i a l extension to obtain the connection between the reduced matrix elements and the polarized angular d i s t r i b u t i o n c o e f f i c i e n t s . The res u l t s are l i s t e d in Table B-1. In t h i s table, the cosines and sines of the phase angles have been omitted for c l a r i t y . I t i s to be understood that where a term i n t and t' occurs, there i s a cos (<|>t-<t>t,) in the expressions for the A^ c o e f f i c i e n t s , and a sin ( 4 > t - < f ' t i ) in the expressions for the B^ c o e f f i c i e n t s . 125 fable B-1 Relations between the Angular Distribution C o e f f i c i e n t s and the Reduced T-matrix Elements * A = 3s* * 3d* + 5p2 * 5f* + 3p? / 0 + 3p* 0 r r l/2 3/2 A^  = 9.487sp - 1.342pd + 9.859df - 7.348sp 3 / 2 - 5.196p d A = 2.5p2 - 1.5d« + 2.857fz • 2.243sd - 1.75pf - 3 . 0 p 2 2 r y P l / 2 + 1.5p2 - 9.487p f - 11,619p p 3/2 3/2 *3/2 A3 = 7.746sf • 8.05pd - 4.382df A^ •= -2.857f2 + 13.997pf B = 4.744sp + 2.682pd - 4.930df - 3.674sp , • 10.392p , d 1 3/2 r3/2 B = -2. 122sd + 1.458pf + 23.718p o / f 2 3/2 B 3 = -2.582sf - 2.683pd + .365df B = -3.499pf * Note that unsubscripted p*s refer to E 2 c a P t u r e » To simplify these expressions, the following replacements are made. s s/-JT d d//T P P//5" f -> f//5" Pl/2 P 1 / 2//3" P3/2 P 3 / 2 / ^ r B-1 126 These substitutions lead to the set of eguations l i s t e d i n Table B-2. The cosines and sines have again been omitted f o r Table B-2 Relations between the angular Distribution C o e f f i c i e n t s and the Reaction amplitudes * A = s 2 * p 2 • d 2 + f 2 + p 2 * p 2 o r r l / 2 *3/2 A^  = 2.450sp - .347pd + 2. 546df - 2.449sp i / 2 - 1.732p 3 / 2d A^  = .5p 2 - .5d 2 • .571f 2 + 1.414sd - .350pf - p 2 / 2 + -5p* / 2 - 3.000p 3 / 2p - 2.450p 3 / 2f A3 = 2.000sf + 2.079pd - 1.131df A^ = -5.71f 2 * 2.799pf B = 1. 225sp + .692pd - 1.273df - 1.225sp 3 / 2 > 2.683p 3 / 2d B 2 = -.737sd + ,291pf + 6.124p 3 / 2f B 3 = -.667sf - .693pd • ,094df B^  = -,7 00pf * Note that unsubscripted p's refer to P 3/ 2 E ^ capture. c l a r i t y . In t h i s table, the t«s and t'»s are now the reaction matrix elements referred to i n Chapter I I I . The t*s and t'*s of Table B-1 are the actual reduced matrix elements and equations B-1 give the connections between the two, 127 Table III-2 has been obtained from Table B-2 simply by dropping terms involving p ^ and p p a r t i a l waves; that i s , terms involving M1 radiation. 128 Appendix C POLARIZED PROTON BEAM ASYMMETRIES In t h i s appendix, the asymmetries measurable with a polarized proton beam w i l l be developed. The two detectors used i n the experiment w i l l be referred to as l e f t (L) and right (R) , with s o l i d angles ^ L and fiR, respectively, the beam polarizations as up (+) and down {+), the yi e l d s as Y, and the amount of beam delivered to the target as Q. Then there are four quantities measured, which are (HA 65) \i - V V 1 + P L i A ) C " 1  YR+ = V W1-- PR+ A ) C " 2  Y L + = \ + " P L + A ) C ~ 3 YR* = QR + V ( 1 + P R + A ) C " 4 In these expressions, Y L + refers to the counts recorded i n the l e f t detector when the incident beam i s polarized up, etc. It has been assumed that the detectors are not sensitive to the polarization and thus A, the analyzing power of the reaction, i s constant throughout. The following further assumptions are now made. The number of protons incident on the target i s assumed to be the 129 sane whether viewed from either the l e f t or r i g h t detector; thus Q = Q for each spin state. I t w i l l be assumed that the beam does not s h i f t when the spin state i s altered, thus ft. = T fi+ for each detector. F i n a l l y , i t w i l l be assumed that the magnitude of the p o l a r i z a t i o n i s the same in the two spin states. Hith these assumptions, equations C-1 to C-4 become YRf " W 1 " P A ) C " 6 YL4- = W l ~ PA) C"7  YR+ = Q A ( 1 + P A ) C " 8 The polarized beam asymmetries follow by taking various combinations of these equations. The analyzing power follows from p - Y l A + _ ( ( ! + PA) 2 which leads to 1 i/I - 1 A = ?jTTT c-10 The charge r a t i o asymmetry follows from Y Y Q 2 •;*Lt Rt_ i?t C-11 Y Y 02" L+ R+ 130 2± = / Li Rt  Q4- J YL-)-YR4-C-12 S i m i l a r l y , the s o l i d angle asymmetry ar i s e s from considering YRt YR+ R^ which gives \ Y L t Y L I "R J *R+ R+ 

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