Study of the p + ³H and n + ³He Final-State Interactions in the Reactions ⁷Li(p,α) and D(³He,p) Citation Lin, Wen Kuan (1969) Study of the p + ³H and n + ³He Final-State Interactions in the Reactions ⁷Li(p,α) and D(³He,p). Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/KPD0-CD06. https://resolver.caltech.edu/CaltechTHESIS:06082017-094850930 Abstract The ɑ-particle energy spectra from the bombardment of 7 Li with 9.1-MeV protons have been obtained at 2.5° ≤ ϴ ɑ ≤ 120°. The high-energy ends of the spectra are interpreted as due to the 1 S p + 3 H final-state interaction through the first excited state of 4 He at 20.06 MeV. The factored-wave-function method is used to deduce the resonance parameters of this state. Consistency in the use of this method is obtained by a PWBA calculation based on the triton-transfer mechanism to account for the forward-peaking in the angular distribution. Coincidence measurements between a-particles and the other charged particles give additional evidence for the 0 + assignment to the state, and indicate that the a + 3 H and a + H final-state interactions are important as the 4 He excitation energy gets higher. To reduce the effects of these final-state interactions, the reaction D( 3 He, p), at a 3 He bombarding energy of 16.5 MeV, has been investigated. The protons emitted from the reaction have been measured at ϴ p = 30° in coincidence with the other charged particles. Angular correlations have been obtained for 6.6 MeV ≤ E p ≤ 8.6 MeV, and compared with a modified Born approximation calculation based on the stripping of 3 He. The angle-energy correlation and the p - 3 H to p - 3 He branching ratio can be reproduced, if Meyerhof's p + 3 H phase shifts and Bransden's n + 3 He phase shifts are used to describe their respective interactions in the final states. In agreement with the reported 0 - state at 21.2 MeV, the p-wave final-state interactions are found to be important in this energy range. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Kavanagh, Ralph William (advisor) Tombrello, Thomas A. (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 May 1969 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Office of Naval Research UNSPECIFIED NSF UNSPECIFIED Record Number: CaltechTHESIS:06082017-094850930 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06082017-094850930 DOI: 10.7907/KPD0-CD06 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10314 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 23 Jun 2017 20:36 Last Modified: 24 Apr 2024 21:57 Thesis Files Preview PDF - Final Version See Usage Policy. 36MB Repository Staff Only: item control page

STUDY OF THE p + 3H AND n + 3He FINAL-STATE INTERACTIONS IN THE REACTIONS 7 Li(p, a) AND D(3He, p) Thesis by Wen Kuan Lin In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1969 (Submitted May 1, 1969) 11 ACKNOWLEDGMENTS I would like to thank the faculty and the staff of Kellogg Radiation Laboratory for the efforts that have made this work possible. In particular, special thanks must go to Professor R. W. Kavanagh who has led the interest of studying these reactions and has given me his continued guidance and encouragement, and also to Professor T. A. Tombrello for his constant interest and his close supervision during the period when Professor Kavanagh was away in France. The author is particularly grateful to Dr. F. Scheibling for his active participation in these experiments and for his introduction to some of the useful computing techniques. The part on the theoreti- cal interpretations has profited from the discussions with Dr. K. Nagatani and Dr. K. Kolltveit. Extra special thanks must go to my wife, Bella, for her patience and confidence and to my mother who is always a source of inspiration. Finally I would like to express my gratitude for the financial support from the California Institute of Technology in the form of tuition scholarships and assistantships. The research project has been supported by the Office of Naval Research and the National Science Foundation. iii ABSTRACT The a-particle energy spectra from the bombardment of 7 Li with 9 . 1-MeV protons have been obtained at 2. 5° ::S 6 a ::S 120°. The high-energy ends of the spectra are interpreted as due to the 1S p + 3H final-state interaction through the first excited state of 4 He at 20. 06 MeV. The factored-wave-function method is used to de- duce the resonance parameters of this state. Consistency in the use of this method is obtained b y a PWBA calculation based on the triton-transfer mechanism the angular distribution. to account for the forward-peaking in Coincidence measurements between a- particles and the other charged particles give additional evidence for the 0 + assignment to the state, and indicate that the a + 3H and a + H final-state interactions are important as the 4 He excitation energy gets higher. To reduce the effects of these final-state in- teractions, the reaction D(3He, p), at a 16. 5 MeV, has been investigated. 3 He bombarding energy of The protons emitted from the reaction have been measured at e other charged particles. = 30° in coincidence with the p Angular correlations have been obtained for 6. 6 Me V ::S E ::S 8. 6 Me V, and compared with a modified Born p approximation calculation based on the stripping of 3He. The angle-energy correlation and the p - 3 H top - 3He branching ratio can be reproduced, if Meyerhof's p + 3 H phase shifts and Bransden's n + 3 He phase shifts are used to describe their respective interactions in the final states. In agreement with the reported 0 state at 21. 2 MeV, the p-wave final-state interactions are found to be important in this energy range. lV TABLE OF CONTENTS PART TITLE PAGE I. INTRODUCTION 1 II. EXPERIMENTAL DISCUSSIONS 7 The Reaction 7 Li(p, a) 9 1. Target Preparation 9 2. Particle Spectra 10 3. Target Thickness 11 4. 7 Li(p, a 0 ) Angular Distribution 12 5. Single a-Particle Energy Spectra 12 6. Coincidence Measurements 14 A. B. III. IV. The Reaction D(3He, p) 17 1. Target Preparation 19 2. Particle Spectra 21 3. Angle Cali bra ti on 21 4. Coincidence Measurements 23 KINEMATICS AND DA TA REDUCTION 26 A. Three -body Kinematics 26 B. Normalization 30 THEORETICAL INTERPRETATION 34 A. Interaction in the Final State of a Reaction 34 B. Resonance Parameters of the First Excited State of ~e C. Triton-transfer Reaction Mechanism in the Reaction 7 Li(p, a) D. 40 45 A Modified Born Approximation Calculation for the Reaction D(3He, p) 53 v PART V. TITLE DISCUSSION OF RESULTS APPENDIX A. PARAMETRIZATION OF THE PHASE SHIFTS APPENDIX B. A SEARCH FOR THE LEAST SQUARE BY ITERATIONS APPENDIX C . PAGE 62 68 74 THE SPIN-ISOSPIN OVERLAPS AND THE SPACE INTEGRALS IN THE MODIFIED BORN APPROXIMATION CALCULATION FOR THE REACTION D(3He, p) 77 REFERENCES 85 TABLES 90 FIGURES 96 1 I. INTRODUCTION The question of the existence of excited states of the a-particle was first raised by Crane, Delsasso, Fowler and Lauritsen (1935) A series of very energetic y-rays with energies as high as 16 MeV were detected in a cloud chamber , when a 7 Li target was bombarded with 1-Me V protons. Those y-rays were attributed to the decay of a-particles produced in excited states. Using a sum-rule argument, Feenberg ( 1936), Bethe and Bacher ( 1936) pointed out that the aparticle may possess excited states. MeV was proposed. An upper energy limit of 20 As was shown later by Austern (1960), these calculations suffer from the fact that the a-particle radius and the nature of nuclear forces were inadequately known in early 1935. When the appropriate corrections are made, the upper limit rises to 50 Me V , indicating that the existence of an excited bound state is not probable. One of the first indications of an excited state of 4He at about 20 MeV was suggested by Frank and Gammel ( 1955), who considered a 1 S resonance at 20.44 MeV to be necessary to explain the energy dependence of the p+ 3 H elastic-scattering cross section. In agree- ment with this, Bergman et al. (1958) also found a 1S resonance at 20. 1 MeV essential to account for the observed departure of the 3 He(n, p) 3H reaction cross section from a l/v law. Stronger evi- dence for such a resonance was later given by Werntz (1962) in his analysis of the neutron energy spectra from the reaction 3 H(d,n) [Lefevre et al., 1962; Poppe et al., 1963]. A strong 1S p+ 3 H inter- action in the final state gives a maximum as required by the data at the high-energy end of the neutron energy spectra. Interpreting the interaction as a resonance, the resonance energy was found at 20. 2 MeV. The reduced widths for p+ 3H and n+ 3He channel were 2 equal to each other, if the channel radius was taken as 3. 0 F. From this Werntz and Brennan (1963) were able to argue that the resonance concerned is a state of definite isobaric spin. The assignment of an isobaric spin to this resonance stimulates various interesting experimental and theoretical efforts and yields useful knowledge about the mass-four system. nance was a T If the reso- = 1 state, the analog 4 Li should be seen also as an s-wave resonance at about 0. 36 MeV (c.m.) in the p+ 3He system and 4 H would be stable against neutron emission by about 0. 18 Me V. The experiments of 4H(i)- ii) 4 H e [Spicer, 1963, Nefkens et al., 19 64 ], 4 H(i)-ii) 4 He':'[Janecke, 1965] and 3H(d,p) 4 H [Rogers et al., 1964] failed to establish any particle stable 4 H state. The phase-shift analysis of the n+ 3 H elastic scattering cross section and polarization data by Tombrello (1966) also indicated that 4 H neither possesses particle-stable states nor low...-lying resonances except for two broad p-wave resonances at higher energies (3. 4 MeV and 5. 1 MeV). Similar results from the p+ 3He elastic scattering were obtained for the 4 Li system. They include Frank and Gammel' s original work, and the phase-shift analyses of Tombrello (1962 , 1965) and Kavanagh and Parker (1966). The two corresponding p-wave r esonances were also found by Tombrello at 4. 7 MeV and 6. 2 MeV in the p+ 3He system. Takin g the coulomb interaction into account, Tombrello's results suggested that the first T = 1 state in 4 He should appear as a p-wave resonance with an excitation energy higher than 24 MeV. The recent phase-shift analyses by Meyerhof and McElearney (1965) and by Balashko , Kurepin and Barit (1966) also confi rm ed the 1 S resonance and showed an increase in p-wave phase shift with energy. One therefore believes that this 1S resonance at 20 MeV = 0 [assumin g isobaric-spin conservation, a direct evidence of the T = 0 assignment for this resonance has been has isobaric spin T 3 reported by Hungerford et al. U 968) from the reaction 4 He( a, a) 4 He':'], and that the T = 1 states may exist at higher excitation energies in pwave states. This speculation was also supported by the theoretical shell-model calculations of deShalit and Walecka ( 1966) and Kramer and Moshinsky ( 1966). The calculation, however, predicts a series of -T = 0 negative-parity states around 22 MeV. Whether the ob- served increase in p-wave phase shift corresponds to a T T = 0 or = 1 state is still a question requiring further investigations. Besides the reaction 3 H(d,n) , the excited states of 4 He were studied by many other three - body reactions. Young and Ohlsen ( 1964), using 6 to l 0 MeV deuteron beams and 3 He gas targets, obtained the proton energy spectra from the mirror reaction 3 He(d, p) over laboratory angles from 14° to 30°. The peak close to the highest-energy end of the spectra was also identified as due to the 0 +resonance of the p+ 3H final-state interaction. foil target and a 3 Using a deuterated polyethylene He beam from a 60-in cyclotron, Donovan (1965) and Parker et al., ( 1965) measured the protons rn coincidence with the 3 H or the 3He. Another excited state in 4 He at 21. 2 Me V of 1. 1 width was found, in addition to the o+ resonance discussed previously. Taking these results, Cerny et al. (1965) were able to resolve a third small peak at 22. 5 MeV in the 3He energy spectra from the reaction 6 Li(p, 3He). The state that gives rise to this peak, however, is not certain, since no such peak was seen also in Cerny's aparticle energy spectra from the 7 Li(p, a) reaction. The present work started with the purpose of re-examining the reaction 7 Li(p, a) by using the tandem electrostatic accelerator and the 61-cm magnetic spectrometer. With better energy resolution and particle identification, it was believed possible to deduce the resonance parameters for the 0 + state and to resolve peaks around 4 22 MeV, if any. The a-particle energy spectra taken at 9.1-MeV bombarding energy at various angles ranging from 2. 5° to 120° show consistent evidence of the existence of the 0 + state in the 4 He system. The obtained resonance parameters and therefore the 1 S p+ 3 H scattering phase shifts are in good agreement with the published values derived from other reactions. The angular distribution of the a-particle group leading to this + 0 state, in contrast to Cerny's results obtained at 43. 7-MeV proton bombarding energy, turns out to be very different from that for the ground state a-particle group. The forward peaking in the angular distribution indicates the importance of the triton-pick-up process. Other peaks which may contribute evidence for the existence of higher excited 4 He states were also seen in the spectra taken at smaller angles. But as the coincidence measurements to be de - scribed have shown, the strong a+ 3 H, a+H and possibly a+N finalstate interactions become important as the energy of the detected a-particles gets smaller. These make the deduction of any useful information about the p+ 3H or n+ 3He interaction very difficult. The coincidence measurements give the angular correlations of the a-particle and other charged particles from the reaction, e.g., between a-particle and 3 He in 7 Li+p-->- a+ 3He+n. When the a-particle momentum was chosen such that there was 20. 014.,.MeV excitation energy in the recoil 4 He system , the correlations obtained can be interpreted as if the recoil 4 He system were decaying isotropically in its center-of-mass system. This strongly confirms the 0 ment of the first excited state of 4 He. + assign- But when the a-particle momentum was chosen such that the excitation energy of the recoil 4 He system was 21. 272 MeV, the protons were found to be very strongly correlated with the a-particles along a certain direction . 5 This corresponds to the situation that the third particle, i.e., the triton, is interacting strongly with the a-particle through the 4. 63MeV excited state of 7 Li. Similarly the a+H final-state interaction via the ground state of 5 Li and possibly the a+N final-state interaction through the ground state of 5 He were also seen in the a- 3 H and a- 3 He correlations, respectively. To reduce the effects of such competing final-state interactions, the reaction D(3He, p) was then investigated. Because the phase shifts of the singlet two-nucleon system stay relatively small at low energies, it was believed that the final-state interactions in the diproton and singlet deuteron system are not as strong as those of interest. Using 16. 5-MeV 3 He bombarding energy, the protons were detected at 30° with respect to the beam. The angular correla- tions of the other charged particles were obtained at proton energies ranging from 6. 6 MeV to 8. 6 MeV in steps of 0. 4 MeV. They all had an axis of symmetry along the momentum _of the re coil 4 He sys tern and suggested the importance of the 3 He stripping reaction mechanism. A modified Born approximation calculation [Yu and Meyerhof, 1966] based on this mechanism was made to estimate the relative amplitudes of producing the final-state interacting pair of particles ins-wave and p-wave states. It was found that both the angular and energy correlations and the ratio of the contributions from p+ 3H to n+ 3He interaction can be reasonably well reproduced, 3 if the p+ H phase shifts of Meyerhof and McElearney (1965) and the n+ 3He phase shifts of Bransden et aL (1956) were used to describe the respective final-state interactions . Very recently, assuming the isobaric spm invariance, Werntz and Meyerhof ( 1968) have made a R-matrix analysis of the 4He system [cf., Meyerhof and Tombrello, 1968]. An energy level diagram 6 shown in Figure l was proved to be consistent with the differential cross section and the neutron polarization data from the reaction 3 H(p, n) 3He. T The 0 and 2 assignments to the second and third = 0 excited states are also consistent with p-wave interactions observed in this work. In Part II the experimental details and the data obtained are discussed. The data reduction along with the kinematics involved is described in Part III. The assumptions of the Watson-Midgal ap- proximation in treating the interaction in final state of a three-body reaction and the applications of such an approximation for dataanalyse s are discussed in Part IV. A summary and discussion of the results is presented in Part V. The appendices are included to supplement the calculations made in Part IV. 7 II. EXPERIMENTAL DISCUSSIONS The 61- cm double -focusing magnetic spectrometer employed in the present experiments is used in conjunction with the ONR-CIT tandem accelerator. It has been described in detail by Groce (1963). With the installation of a nuclear magnetic resonance (NMR) magnetometer, the spectrometer was carefully calibrated by McNally (1966) in his Q-value measurements. When a particle of mass M and of charge Z passes through the magnet, its kinetic energy E, as a function of the magnetometer frequency f is given by M E =k f2 Z 2 _p_ ( 1 M E ), 2 2Mc where M is the proton mass and k is a parameter to be determined p experimentally. Because of the location of the magnetometer and the dependence of the magnetic field profile on the field strength, k was found to increase by 0. 88% as the frequency changes from 20 MHZ to 44 MHZ. A conversion table from frequency to energy, con- structed by using two measured k values for two separate frequency ranges, was used in all the energy measurements of these expe riments. A correction to this table, which is less than 30 keV for both protons and a-particles, was made at the frequencies around 33. 8 MHZ. This is the frequency where the division into two sepa- rate frequency ranges occurs. Disregarding the relativistic correction, particles of the same kinetic energy and of the same Z 2 /M will correspond to one magnetometer frequency. Another measurement, usually the energy loss of the particle in certain stopping material, is needed to remove this ambiguity for proper particle identification. This is done by placing either a ~E-counter or an appropriate stopping foil plus an 8 E-counter on the focal plane of the magnetic spectrome ter. As the energy loss is roughly inversely proportional to the particle kinetic energy, the conditions, such as the bias on the 6E-counter or the stopping-foil thickness, must be suitably adjusted as the energy changes from one range to another. To make these adjustments more manageable, a single surface-barrier counter (200 mm 2 , 140µ thick at 50-volts maximum bias) was used . Using thin nickel foil as the stopping material, a-particles of energy as low as 900 keV could be separated out from the protons. As seen from Fi g ure 2, the measurement of low energ y a-particles by the spectrometer was limited by the presence of continuously distributed pulses below 600 ke V. From the width of proton group, the over all ele c- tronic noise was estimated to be less than 250 keV. Those continu- ously distributed pulses probably come from the randomly scattered particles by the wall of the spectrometer. The energy resolution oE / E of the spectrometer, controlled by a slit of adjustable size in front of the counter, was set either at 1. 11 % or 0. 56%. The entrance slits, target chamber, and target holder used have been discussed in detail in Groce 1 s work, and no further description will be given he re. Each of the surface-barrier counters used in the experiments was connected through a cable of minimum length to a TENNELEC Model 1 OOA low noise preamplifier . The pulses from the preampli- fier were amplified by an ORTEC model 410 linear amplifier and were then analyzed by a RIDL 400-channel analyzer. After each run the memory of the analyzer was read onto paper tape for subsequent determination of yields from the peak areas. 9 A. The Reaction 7 Li(p, a). In order to investigate the effects of p+ 3H and n+ 3 He final- state interactions on the a-particle energy spectra, a high bombarding energy is desirable. The a-particles can then escape from the recoil pair of particles with large relative velocity without interacting with either particle of the pair. Without knowing anything about the reaction mechanism beforehand, one wishes, however, to 7 choose a beam energy such that the compound system of the target Li nucleus plus the incident proton may have a resonance (for increased yield). Thus the proton bombarding energy was chosen to be 9. 1 MeV, corresponding to the 2 + 25. 2-MeV excited state of 8 Be [Lauritsen and Ajzenberg-Selove, 1966]. The 9. 1-MeV proton beam was obtained from ONR-CIT tandem accelerator. A negative proton beam - 20 µA was extracted from the negative ion source and was stripped to a positively charged beam at the center terminal of the tandem. a 86 . 3-cm uniform 90° -rnagnet. It was then analyzed by When the object and image slits were set at 3. 81 mm, and the beam-defining slits in front of the target chamber were set at 1. 53 mm along the horizontal and vertical direction, a beam of - 1 µA was normally obtainable on target. In the target chamber, there was an additional surface-barrier counter. Except in the coincidence measurements, it was fixed at 145° w i th respect to the beam direction and measured the flux x target density of the reaction. An accurate integration was thus not re- quired except for the runs during which the target thickness was being measured. 1. Tar g et preparation. The 7 Li targets were prepared from 99. 99% 7 Li enriched 10 metal in the target-chamber furnace illustrated by Groce ( 1963). The lithium metal in a tantalum boat was preheated to eliminate the kerosene in which the lithium had been stored. As the current through the boat increases , the pressure of the targ e t chamber will rise suddenly when one of the compounds in the target material starts to evaporate . A compound, presumedly the lithium hydroxid e which has a boiling point considerably lower than that of the lithium metal, was found to evaporate first. The preheating process is finished, if the pressure stays at norma)_ value - 1. 5 x 1 o- 6 mm Hg as long as the currer,t is kept below the value at which the lithium metal would start to evaporate . The gold backin g foils - 80µg/cm 2 , mounted on the target hold e r are now lowered down to the level of 'the tantalum boat. To evaporate the lithium onto the foil, the temperature should be built up gradually to avoid breaking the foils. From the time duration of the evaporation, or from the color that the foils appeared, the amount of the lithium deposited on the foils is roughly known. It can be checked by measuring the yields from the reaction 7 Li(p, u 0 ) with the monitor-counter in the target chamber. . Additional evaporations can be done easily without opening the vacuum system. 2. Particle spe ctra. The monitor-counter (50 mm 2 , 300µ thick at 85-volts maxi- mum bias) located in the target chamber provided quick surveys of the particles emitted from the reaction. obtained at 60° and 90°. Figure 3 shows the spectra In spite of the elaborate precautions taken in the target preparation , the oxygen and carbon contaminations were not avoided. 11 In the energy region where the contribution due to the reaction 7 Li(p, afHe:::' is expected, one finds the presence of strong competing processes, such as 16 0(p, p'), 12 7 C(p, p 1 ), possibly some other three-body reactions. Li(p, d 0 ) and The usefulness of ap- plying the magnetic spectrometer to identify the particle species is manifest. 3. Tar getthickness. In the first attempt to measure the target thickness, a gold foil - 80µg/cm 2 uniformly coated with a CaF 2 layer - 20 tJ.g/cm 2 was prepared by vacuum-evaporation. The thickness of the 7 Li, evapora- ted later on top of the CaF 2 layer in the target chamber, was determined by measuring the energy loss in 7 Li of the 3. 85-MeV a-particles produced from the reaction 19 F(p, a') with 5-MeV protons as the bombarding particles. The energy loss was then converted into the number of 7 Li atoms per cm 2 by using the atomic stopping cross sections given by Dernirlio glu and Whaling ( 1962). The results are shown in Figure 4 and the differential cross section of 7Li(p, a 0 ) at 30° was found to be 1. 39 ± 0. 35 mb/sr. This is considerably lower than the 2. 33 ± 0. 43 mb/sr calculated from the coefficients of the Legendre polynomials published by Mani et al. (1964). The uncer- tainty in the oxygen and carbon contamination leads to an underestimation in the actual average stopping cross section. A target of some lithium compound with known chemical composition should be preferable. Also by vacuum-evaporation a LiF target - 80tJ.g/cm 2 on gold backing was then prepared. Its thickness was determined by measuring the energy loss of a 9. 1-Me V aparticle beam from the tandem accelerator. Without including the errors of scaling the proton stopping cross section to that of the 12 a-particle and the assumption on the LiF chemical composition, the differential cross section of 7 Li(p,a 0 ) at 30° was found to be 1. 88 ± O. 08 mb/sr. This value will be taken to convert the measured rela- tive differential cross sections into the absolute ones. 4. 7 Li(p, a 0 ) angular distribution. Be sides for checking the normalizations, the 7 Li( p, a 0 ) angu- lar distribution is interesting for a comparison with that from the reaction 7 Li(p, a 1 ). Since the first excited state of the a-particle was found to have the same spin and parity as the ground state of the a-particle , it was expected that the two angular distributions should bear some re semblance [Cerny et al. , 196 5]. As would be anticipated for a reaction involving two identical bosons in the final state, the angular distribution shown in Figure 6 has a 9 0° symmetry in the center-of-mass system. The data obtained at higher bom- barding energies by Maxson ( 1962) are also included for comparison. The shapes and the absolute values of the angular distributions for these bombarding energies appear approximately the same. This confirms the direct triton-pick-up reaction mechanism that Maxson has discussed in his analyses. 5. Single a-particle energy spectra. In the three-body reaction 7 Li + p--+- a+ p + 3H, the excitation energy of the p+ 3 H system is fixed, if an a-particle of known energy is detected at a given angle. As a typical procedure in obtaining an a-particle energy spectrum, the spectrometer frequency was first set to a value such that an associated 4 He of 20. 0-MeV excitation energy is expected to be formed. The protons having the same en- ergy and the same Z 2 /Mas the a-particles were separated out by varying the bias voltage of the counter or the thickness of the stop- 13 ping foil. For further ass urance of detecting the correct group of particles, the spectrometer frequency was then increased step by step . If the group of particles was due to the a-particles emitted from the reaction, the counting rate of the group should reduce to zero when the magnetic field strength of the spectrometer reaches a value such that the corresponding excitation energy of the recoil 4 He system crosses its p+ 3H threshold from above. Once the a-particles group had been established, the change in pulse height of the group could be followed as the spectrometer frequency decreased when the lower-energy part of the spectra were taken . Some adjustments on either the bias voltage or the stopping foil thickness were necessary to keep the proton group away from the a-particle group. The yield from 7 Li(p, a 0 ) was measured at 145° with the monitor-counter during each run for normalizing the flux x density. At the beginning it was considered to be interesting to investigate the a-particle energy spectra over a wider energy range. of these spectra are shown in Figure 7. Two When the phase space factor has been taken out [cf. , equation (5) in Part III], the transition probability is plotted out in Figure 8 as a function of the 4 He excitation energies . The contribution due to the second excited state of 4 He app e ars as a peak in the spectra near 21. 2 MeV in agreement with the value that Parker~ al., (1965) and Cerny .s:..!_ al., (1965) have found. This becomes less obvious and is masked by the contributions from other final-state interacting pairs at larger angles . To unfold any useful information about the second excited state would require a rather complicated analysis [Morinigo, 1963; Bacher, 1966]. 14 Limiting the interest to study of the first excited state of 4 He, only the high-energy ends of the spectra were taken in the later stage of the experiment. It includes the 4 He excitation energies from the p+ 3H threshold, 19. 814 MeV, up to 21 MeV. Figure 9 shows the spectra obtained at the laboratory angles from 10° to 120° . 6. Coincidence measurements. For a three-body reaction 1 + 2 -+ 3 + 4 + 5, nine variables are required to describe the m o mentum vectors of all particles in the final state . The conservation of total energy and momentum reduces this number to five. The range that these five independent variables are allowed is ordinarily referred to as the phase space. Only three out of the five variables were measured with known resolutions in the measurements of energy spectra 'described in the previous section. Except for the pair (4 + 5), the pair excitation energies were not fixed. The coincidence measurements go one step further to measure these energies and consequently are limited to a smaller region of phase space. As will be explained in Part III, this is done by measuring the direction along which the particle 4 is emitted. If the pair interactions of (3 + 4) and (3 + 5) in the final state of the reaction are not strong as compared with that of the pair (4 + 5) over the kinematically allowed phase space region, the angular correlation of particles 3 and 4 should include the same information about the spin, parity and some other characters of the state that the pair interaction (4 + 5) leads to. From the single a-particle energy spectra of the reaction 7 Li(p, a) 4 He':', the p+ 3H final-state interaction is seen definitely to imply a 4 He first excited state. It would be interesting to check if the state prepared in this way agrees with the 0 + spin-parity assignment from other sources of experimental data. 15 Keeping the incident proton energy still at 9. 1 Me V, the aparticle s were detected also by the magnetic spectrometer at 30° rn coincidence with other charged particles of the reaction products. With 68 = 1 °, o<I> = 4° and oE/E = 1. 11 % , the spectrometer was set to detect 5-Me V a-particles. Its exact field strength, depending on the thickness of backing foil and that of target, was found and fixed at the value that maximized the a-particles counting rate. Other charged particles were detected by the monitor-counter in the target chamber. Its angular apertures were set at 3. 8° and 15. 5° respectively along the polar and azimuthal directions. Since the protons and tritons were detected around -110°, the counting rate was not high, and a slow coincidence circuitry was sufficient. Figure l 0 shows the electronic arrangements. Due to the transit time of the a-particles through the vacuum box of the spectrometer, the pulses from the counter in the target chamber were delayed by O. 2 µs before entering the coincidence circuit. The ORTEC Model 420 timing single-channel analyzer, having a 10-turn control to delay its outputs up to 1 µs, provides this amount of delay. The coincidence resolving time of the ORTEC Model 409 linear gate and slow coincidence is equal to 1 µs (the sum of the two input-pulse widths). The coincidence output gives enable pulses to open the linear gate through which the pulses from the counter in the target . chamber were directly put into the ADC of a multi-channel analyzer. A 1. 7-µs delay to these pulses was necessary to assure the gate to be opened at the right time. The coincidence outputs were also used to gate another multi-channel analyzer in setting the window of the single-channel analyzer for the pulses from the magnetic spectrometer. 16 The obtained coincidence spectra are shown in Figure 11. They were taken with chamber. 7 Li target facing the counter in the target The a-particles have to pass through the gold backing and suffer the spreadings in both the energy and the direction of motion. The resolution in the coincidence spectra was found to improve a little , if the target was rotated around by 180°. But the kinematic dE/ d6 spreading, the most important one, cannot be improved without cutting down the coincidence counting rate. No at- tempt, the ref ore, was tried to separate out the tritons and protons in the coincidence spectra. A single spectrum was taken right after each of the coincidence runs. From the counts in the coincidence spectrum under the proton peak from the elastic scattering by the gold backing, the randoms per channel can be estimated. After these randoms were subtracted, the sum of coincidences due to protons and tritons is shown in Figure 12. The curve is the total coincidence efficiency predicted by assuming an isotropic decay of the 4 He system. · agreement gives additional evidence to support t h e The o+ assignment · for the first excited state of 4 He . When the spectrometer is set to detect a 4. 1-Me V u-particle, more energies are then available for the decay of the recoil 4 He system. It is now excited above the n+ 3He threshold, so that among the reaction products there are also neutrons and 3He 's. Carbon foils -~ 20 fJ,g/ cm 2 were tried to reduce the spreadings of both the energy and angle due to the target backing. In this case larger angular apertures for the counter in the target chamber are allowable, because dE/de is smaller. Keeping the same width in azimuthal di- rection, the width along the polar direction was increased by a factor of two. Figure 13 shows some of the coincidence spectra. The co- 17 incidence data after subtracting the randoms are plotted in Figure 14. By detecting an a-particle at a certain momentum , the excitation energy of the pair (p+ 3H) or (n+ 3He) is fixed r e gardless of where the proton or 3H (neutron or 3He) is detected. The rela- tive energy of the other two pairs of particles depe nds symmetrically on where particles other than the a-particle are d e tected [cf., Part III]. It has the same value on a cone determined by an axis of revolution around the 4 He recoil axis in the center-of-mass of the entire system. Because of the experimental arrangement, the particle 4 is always detected almost opposite to the direction of motion of the particle 3, i.e., the a-particle, particle 4 escapes from the a.particle with higher relative velocity than the und e tected particle 5 does. Since the final-state interaction is stron ger for the pair of particles having lower relative velocity, the a-H coincidences are enhanced by the a+ 3H final-state interaction through the 4. 63MeV excited state in 7 Li at the two angles determined by the intersections of the cone and the scattering plane. Similarly the effect of the a+H final-state interaction through the_ ground state of 5 Li is also seen in the a- 3H coincidences. The enhancement of this kind, however , is not so obvious in the case of the a.- 3 He coincidences shown in Figure 14. B. The Reaction D(3He, p). From the coincidence measurements described in Section A6 , it was found that the strong final-state interaction of pair (3 + 5) may come in to mask that of the pair (4 + 5) which is the primary 18 interest of this work. To overcome this difficulty, one may try to change the beam energy or the angle where the particle 3 is detected. Hopefully a phase~space region may be found, such that particle 3 won't interact as strongly as particle 4 with the particle 5 in the final state of the reaction. Instead of trying to search for such a phase space region, the reaction D(3He, p) was investigated . The interactions that may come in to mask the effects of the p+ 3H and n+ 3He final-state interactions are respectively those for the diproton and the singlet deuteron systems. As mentioned in Part I, this reaction was studied before by Donovan and Parker ( 1965) [cf., Ziirmuhle, 1965]. Only the po- sitions of the first two excited states were reported. It would be interesting to look into the angular correlations of the decaying particles to study some other properties of the states. From the standpoint of studying the reaction mechanism, Yu and Meyerhof (1966) have analyzed the single-nucleon energy spectra from the reactions 3He(d, p) and 3H(d, n). From their modified Born approxi- mation calculations, they concluded that the singles only cannot distinguish between the different break-up modes and suggested that the coincidence measurements would help to resolve this difficulty. To separate different groups of particles from one another in the coincidence spectr a , and also for the same reason described at the beginning of Section A , it is better to use the 3 He beam energy as high as possible. However it was found that the tandem accel- erator can give a more stable beam by using 5. 5 million volts as the terminal voltage rather than 6 million volts (its nominal maximum). 19 A singly charged negative 3He beam was extracted from the negative ion source and was stripped to a double positive charge at the center terminal of the tandem. Using the same sizes of ob- ject, image and beam defining slits as described in Section A of this Part, a 16. 5-MeV 3He beam of 5 to 20 nA was obtained on target. 1. Target preparation. In Donovan and Parker's experiment, a thin deuterated polyethylene foil target - 1 mg/cm 2 was used-. A solid target is more convenient than a gas target, since for a coincidence measurement a well defined beam spot is essential. At the time when the present experiment was planned, the deuterated polyethylene was not commercially available. A compound called the deuter- ated dotriacontane, suggested and supplied by its manufacturer (Merck, Sharp and Dohme of Canada), was used for the target preparations. To prevent loss of dotriacontane which would be incurred during vacuum evaporation, it was instead put in carbon tetrachloride, stirred until it dissolved completely, and then the solution was carefully transferred onto a mounted backing foil with a medicine dropper. When the carbon tetrachloride had dried, the target was ready to be used. Using the heavier particle as projectile, reaction products such as 3H' s and 3He' s are emitted in forward directions. It is im- portant to choose a low-Z material as target backing in order to reduce the contribution to the counting rate from the Rutherford scattering. On the other hand, the melting point of the material 20 is only 68° C, and for better target stability against the heat deposited as the beam particles go through , a metal with high thermal conductivity is preferred. Taking those factors into consideration, copper - 50 µg/ cm 2 was chosen. The deuterated dotriacontane, CD 3 (CD 2 h 0 CD 3 , was indicated by the manufacturer to be 99. 7 atom% in D by mass analysis. Use of the proton yields detected by spectrometer to normalize the coincidence spectrum has the shortcomin~ that the target may deteriorate and pick up carbon during the run. Likely the protons could · . iz C (3 He, p )14N':' . come from the react10n In the energy range of inter- est, many excited states of 14N [Aj ze nberg-Selove and Lauritsen, 1959] may contribute to the proton counts. The total number of pro- ton counts is therefore proportional to some linear combination of deuterium and carbon atoms in unit area of the target, while the coincidence counting rate is proportional only to the number of deuterium atoms per unit area. It is all right to use the proton counts as normal- izer, if the chemical composition of the target material does not change [cf., Section B, Part III] . To check if the chemical composition of the target material changes, another surface-barrier counter was set up at 60° in the target chamber in the last day of this experiment. It detected the deuterons from the elastic scattering D(3He, d) 3He and its yields should be proportional to the number of deuterium atoms per unit area. The number of protons counted by the spectrometer divided by the number of deuterons counted by the additional counter was plotted in Figure 15 against the total integrated charge on target. The ratio, within a 10% deviation from its average, is reasonably constant for the integrated charge less than 70 µC. Most of the 21 targets broke before they had accumulated this much of a charge. Figure 15 also shows the ratio plotted out as a function of proton energies. These numbers were used to normalize the coincidence runs taken at different proton energies including an additional 10% deviation. 2. Particle spectra. Due to the presence of carbon nuclei in the target material, the spectra shown in Figure 16 reflect a rather complicated structure. The processes such as 12 C(3He, 3He) 12 C ':', 12 C(3He,p) 14 N':' and 12 C(3He,d) 13N':' are possible. 12 C(3He,a) 11 C':', All of these recoil nuclei possess many excited states, and no attempt was mad e to identify each of these possible contributions. detecting a The limiting angle of 3 He from the D+ 3He elastic scattering is 42°, therefore no such group was seen in the particle spectrum at 46°. The recoiled deuteron from the elastic scattering was not obvious at 26 ° and smaller angles. 3. Angle calibration. In the coincidence measurement, one of the counters was set up to detect a particle at certain fixed momentum, while the other counter was moved around. The angular correlation obtained in this way is essentially an angular distribution of the breakup of the recoil system . The conversion to the recoil center-of-mass system will be explained in Section A, Part IlI. As will be noticed, both the cross-section conversion factor and the recoil center-ofmass angle are very rapidly varying functions of the laboratory angle es pe ciall y when it a ppr oache s one of the limiting angles. 22 The precision of knowledge of the angles was therefore an important consideration. The polar angle of the counter in the target chamber could be set to an accuracy of ±0.1° [Groce, 1963]. It was controlled by a dial (36° per turn) attached to the lucite cover of the target chamber. Since the target chamber rotates with the spectrometer, the angle that a particular dial reading indicates also changes with the position of the spectrometer. Originally the dial was so adjusted that it read 90°, when the counter was actually 90° with respect to the beam and the spectrometer was set at 0°. The precision in meas- uring the polar angle of the counter in the target chamber, therefore depends on how well the lucite cover can be reproducibly set to the correct position. The following method provides a check to this question. For an elastic-scattering process such as D + 3He ~ d + 3He, there exists a unique pair of angles on the scattering plane given by e c = ± tan- 1 where both the deuteron and 3He can be detected in coincidence with each other. Unless the spectrometer has been set ate , the angles c that the counter in the target chamber should be set to detect deuteron in coincidence with 3He in spectrometer and vice versa are different. By measuring this difference, the absolute angle of either of the counters can be calculated. Since e and the difference as c a function of the spectrometer angles shown in Figure 17 are independent of the energies, this calibration requires only a measurement of the difference in angles. 23 To begin with, the spectrometer with 68 = O. 2°, o<I? = 4° and oE/E = 1. 11 % was fixed at 37. 75° according its reading. The NMR magnetometer frequency was respectively set to detect 3He's and deuterons from the elastic-scattering process. Figure 17 also shows the number of coincidences versus the dial readings, corresponding to different angles of the counter in the target chamber. , the 3 He- and c deuteron-coincidences almost peaked at the same position. The Because the spectrometer was so closely set to 8 small difference in the positions of centroids was calculated to be 2.4 ± O. 5 units of the dial reading (0. 24 ± 0. 05°). The actual angle of the spectrometer was then found to be 37. 62 ± 0. 04°, and a dial reading of 300 would imply that the counter in the target chamber was set at 38. 11 ± 0. 08°. When the spectrometer was changed to 30° for the cornc1dence measurements of the reaction D(3He, p), the same reading of 300 became an angle of 45. 73 ± 0. 09°, which would be otherwise taken as 45° if the lucite cover of the target chamber was assumed to be set at the correct position. 4. Coincidence measurements. The protons produced in the reaction D(3He, p) were de - tected by the magnetic spectrometer at 30°. To separate the a-particles that may be present from the protons, a 0. 25-mm thick aluminum sheet was put in front of the counter. The other charged particles from the reaction, namely protons, tritons and 3 He 's, we re detected in the target chamber by a surface-barrier counter. This counter could be set on the plane determined by the beam direction and the center of the spectrometer entrance 24 slits within ±0. 5mm [Groce, 1963]. The angular ap e rtures were 2° and 8° respectively along the polar and azimuthal direction. As discussed in a previous section, the counter in the target chamber had to be set at quite forward angles to permit coincidence counts. The Rutherford scattering from both the target and the backing then contribute most of the counting rate, and a fast-slow coincidence system [cf., Figure 18] was used to reduce the randoms. As in the coincidence measurements for the 7 Li(p, a) reaction, the pulses from the counter in the target chamber were delayed by 80ns before going into the fast-co i ncidence circuit to take account of the proton transit time in the vacuum box of the spectrometer . The fast-coincidence output gives the enable pulses to open the linear gate for those pulses directly from the linear amplifier for the c ounter in the target chamber (l.5µs delayed). The outputs from the linear gate were then put into a multi-channel analyzer which again was promptly gated by the stretched fast-coincidence outputs. The delays and the fast-coincidence resolving time (l lOns) were checked by a coincidence measurement of the elastic scattering D(3He, d) 3He at the beginning of each running day. In all the coincidence runs, the resolutions of the magnetic spectrometer were set at 68 = 1 °, o<I> = 4° and oE/ E = 1. 11 %. By changing the detected proton energy with the counter in the target chamber fixed at .,. 20°, one has the coincidence spectra for the energy-correlation shown in Figure 19 . Fixing the proton energy at 7. 8 Me V instead, those obtained f_or the angular correlation are shown in Figure 20. 3 3 The p-H correlation for the reaction D + He .-. p + H + H can be converted to a p - 3 H correlation, since the proton and 25 triton are emitted oppositely in the center-of-mass of the recoil "He system. By using the formula to be described in the next 3 part, the p - 3 H and p - He correlations shown in Figure 21 were normalized and converted to the recoil center-of-mass system. The randoms were subtracted in the same manner as described in Section A6. If the final-state interaction of pair (3 + 5), i . e., the pair (p + H) or (p + N), was strong, its effect should show, de pending on the energy of proton detected by the spectrometer, in 3 3 the p- H or p- He angular correlation around eLAB = 30° (8RCM = - 70°) where the excitation energy of the two-nucleon system was smaller. No enhancement, however, was seen at these positions. When the angular correlations in the recoil center-of-mass = 7. 8 Me V were analyzed by a least-square fit to an p even order of Legendre polynomials, the p- 3H to p- 3He branching system at E ratio was found to be 7. 86 ± 0. 69. The other interesting feature about the correlations is that they all appear to have an axis of symmetry along the 4He recoil direction. As will be discussed in Section D, Part IV, it is possible to use this symmetric property to show that the mechanism representing the stripping of incident 3 He is the dominant process of the reaction .' 26 III. A. KINEMATICS AND DATA REDUCTION Three-body Kinematics. If T . is the transition matrix element for a process from fl a certain initial state i to a certain final state f, then by the famous golden rule the transition rate is given by Prob./Sec. I 1z dE, dN = 2"IT ti Tfi where dN/dE is the density of final states per unit energy interval. It is expressed as 1 dE for an n-particle final state. A more symmetrical form can be obtained [Feynman, 1962], if one adds a factor of _,. n l dP (z1Tti~ 3 (21Tti) 3 o(:; j Pj) =1 n and replaces l / dE by o(E - .l E.), where ":fr and E are the total . 1 J J= momentum and energy available for the transition . Defined by the experimental resolutions , the transition leads only to a finite part of the phase space region AT. Let the corresponding transition rate be denoted by Awif' then 27 Awif = s ~1T n n ~Pj)o(E 3 2 1Tfil (21T11) o(1t - AT ~ dP m ( 21T11) 3 j=l If the initial continuum-state wave function is normalized to one particle per unit volume, the flux is numerically equal to the relative velocity of the two particles in the initial state, and the differential cross section is just 0.cr = A~f /flux. For a reaction leading to a three-particle final state, the integration can be carried out easily. volved be labelled as l barding particle, 11 Let the particles in- + 2 _,,. 3 + 4 + 5, where 11 1 11 is the born- 2 11 represents the target and 11 3 11 is taken to be the particle detected in the magnetic spectrometer. The differential cross section for the three-body reaction is now given as 1 SI Tf. l 2 ~ .;:2: ~ ~ 4- ~ ~ m 1 + 2m 2 + 1 6(..t-'1 + ..1:-'z -..t-'3 - .t-'4 - Ps)o( 2 AT (1) Q In this expression, non-relativistic energy, i . e., Ei = Pi 2 /2mi + m.c 2 , is used. l v 12 is the relative velocity of the particles 1 and 2, and Q = (m 1 + m 2 the reaction . - m3 - m4 .:. m 5 )c 2 is the Q-value for After having integrated over P5 and P 4 , the ex- pression (1) becomes (2a) 28 i\ ) = 0 2 (i12 - P3 - with 2m 5 (2b) Here ~E. and ~ri. are J J respectively the energy and angle resolutions of the detector that to represent the conservation of energy. particle j enters, and P4 is the unit vector in the direction of i\. rt lJ.. and The total and relative momenta of the particles i and j , qlJ.., are ~ (pi -2) = _,, 1T . . = LJ :P.l + :P .. J m.· m. l ~ J q.. = m.+m . lJ l J m. l ~ m . . v ... lJ lJ J In the study of final-state interactions, the relative momenta q\ and q45 are found more convenient to be used. The vector q., defined 1 as ...,. q. l m.(m.+mk) pi J = m.+m.+mk (m . 1 1 J 1 where i, j and k are in cyclic order of (3, 4, 5), is the relative momentum of particle i with respect to the recoil pair of particles (j + k). In terms of these momenta, the expressions corresponding to (2a) and (2b) are written as ~CT and = 2Tim 1g 1i ql2 1 (21T1i)6 2 ~E3 ~r23 ~r245 m3 P3JTfi1 m4s q45, s.L + Q - -9.iL 2m12 2m45 m3 +mi +ms q32 2m 3 (m4 +ms) = 0, (3a) (3b) 29 where Ar2 45 is the angular resolution Ar2 4 seen in the recoil centerof-mass system of particles 4 and 5. The ratio Ar24 / Ar24s is ob- tained by equating the expressions (2a) and (3a), and is given by ~ Ar2 45 ~A ~lZ - . .J:-' ~I = ~I 2 p 4 p 4 • ( lT 3) • P ms ( 4) 4 This is just the result that one would obtain in transforming an elementary solid angle from one system to another, the vector (ii12 - F3} / (m 4 +ms) is the relative velocity of the two systems. ~ In the laboratory system, the target is at rest, thus P 2 = 0, .;;."IZ = P1 and Ci'12 / m12 = P1 / m1. If only the single spectrum of particle 3 is measured, such as in the measurements of a-particle energy spectra discussed in Section A5, Part II, the differential cross section has to be integrated over all the directions that particle 4 may be emitted. That is ( 5) " 4 is also fixed. In the coincidence measurements, P The relative energies for all pairs of particles in the final state, depending on the squares of the relative momenta, __m_..1.._Tf_...12.......___ + m3+m4+m5 (6a) ( 6b) 30 _k] and ( 6c) ms are known. As is seen from the expression (3b), q 45 is fixed by q 3 and is independent of where the particle 4 is detected. both But q34 and q35 depend linearly on q45 , and each of them obtains the same value on a cone determined by an axis of revolution q3 and t h e angle between "q3 and "q45 • For a specific example, the velocity vector diagram of the reaction p + 7 Li-+ a.+ 3H +pat E Figure 22. = 9. 1 MeV is shown in p A 4.1-MeV a.-particle was detected at 30° with res- pect to the beam axis, while the recoil system of (p + 3H) was moving with the same speed in the opposite direction in the centerof-mass system. This recoil direction was indicated by q 3 • The proton and triton, i.e., the particles 4 and 5, were emitted oppositely in their center-of-mass system along the direction q45 • As long as the protons were detected on the cone determined by q3 and q45 , the relative velocities v 34 and v 35 , thus the relative energies of the corresponding pairs (3 + 4) and (3 + 5), remained the same. In the discussions of Section A7, Part II, the protons and the a.-particles were found to be very strongly correlated when the protons were detected at 26° with respect to the recoil axis q3 . Because the relative energy of the pair (a. + 3H) was equal to 1. 96 MeV, the enhancement in the angular correlation was attributed to the final-state interaction of the pair through the 4. 63-MeV excited state of 7 Li. B. Normalization. In this work, there are two major types of experimental 31 data. Those presented in Figures 7, 9 and l 5b are the energy spectra. The data which were expressed in terms of the particle- particle correlations belong to the second type, and are shown in Figures 12, 14 and 21. The normalizations made in reducing the data are explained in this section. The yield of the magnetic spectrometer is expressed in terms of the spectrometer resolutions and the differential cross sections d 2cr/dE 3 dS1 3 by 2 + _d__cr_c__ AN c) (7) dE 3 dr2 3 ANt where AN and AN are the numbers of the target nuclei and the c contaminating nuclei per unit area. ANB is the total number of t beam particles that have struck the target. For the reaction 7 Li(p, a.), either d 2cr /dE 3 dr2 3 or AN / ANt c c was small, so that the second term in the expression (7) was neglected as compared with the first term. With the 7 Li(p, a. 0 } yield Y monitored at 145° during the same run, Y was then con. mon spe verted to differential cross section in a unit of mb/sr-MeV as y 90 spe x 0. 86 mb/sr-MeV, = 2.643x E xy 3 mon where the first constant is the ratio of solid angle of the monitor counter to that of the spectrometer, and AE 3 = E 3 /90 is the energy resolution of the spectrometer . The last constant is the 7 Li(p,a. 0 } differential cross section at 145°. From the measurements of the 32 target thickness and the 7 Li(p, a-()) angular distribution described in Sections A3 and A4, Part II, it was found to be 0. 86 ± O. 09 mb/sr. The coincidence yield, with the randoms subtracted, is related to the third order differential cross section derived from the expression (2a) by y . ( 8) coin The number of gates from the spectrometer, given also by Y spe in the expression (7), was a measure of the flux x density, i.e., ~NB x ~Nt. 7 For the coincidence measurements on the reaction Li(p, a), the coincidence counts were normalized to l 000 gates of the a-particles, and are plotted out versus the laboratory angles in Figures 12 and 14. The complications in the reaction D(3He, p), due to the possibilities of the carbon and oxygen deposition and the deterioration in chemical compositions of the deuterated dotriacontane during the run, have been discussed in Section Bl, Part II. A direct measurement of ~NB x ~Nt was made by using a third counter to monitor the D(3He, d) 3He elastic scattering yield at 60° in the last day of t his experiment. Since most of the coincidence spectra were obtained without this counter, the number of gates was used to normalize the corresponding coincidence spectrum . The number of protons counted by the spectrometer normalized to l 0 4 of those deuterons from the third counter, plotted in Figure l Sb , was used to account for the energy-dependent factor of the expression (7). Denoting this factor by N{E 3 ) , the relative yield versus the recoil 33 center-of-mass angles given in Figure 21 is expressed in terms of the various yields as Relative yield = O. 0796 x ~~ x N(E 3) x :coin x spe where the first constant was chosen such that the product of the first three factors is unity for the coincidence spectra taken at E 3 = 7. 8 Me V. The ratio t::;.0. 4 / t::;.0. 45 , given by the expression ( 4), transforms the differential cross section from the laboratory system to the recoil center-of-mass system. The recoil 4He direction, i.e. , ~ 4 5 , was used as an axis of reference. 34 IV. THEORETICAL INTERPRETATION The discussions carried on in Part II and Part III relate the experimental data to some appropriate differential cross sections. Besides the kinematic factors, these differential cross sections contain the squared modulus of some transition matrix elements. In this part some assumptions will be made to construct an approximate form for the transition matrix element T fi. The object is to see if the data could be understood in terms of these assumptions. Some parameters concerning the properties of the final-state interactions will be discussed. A. Interaction in the Final State of a Reaction. The interaction between particles produced as a result of a certain reaction may have a sizable effect on their distributions in energy and in angle. This effect becomes particularly notice- able when the reaction proceeds into a phase space region where the relative velocity of the interacting pair of particles is small. In case of a three-body reaction, this part of the phase space region is usually reached by detecting the particle 3 near its maxirnum possible energy. Approximate treatments of final-state interactions are found in the works by Migdal (1955), ·Watson (1952), Goldberger (1964) and Gillespie (1964). Here only the basic idea of the so- called factored-wave-function method or the Watson-Migdal approximation will be discussed. By virtue of the short range nuclear interaction, the particles in the final state of a reaction are created in the vicinity 35 of one another, and would escape from there in forms of plane waves, if there were no final-state interactions acting among themselves. Because of these interactions, the wave function of the relative motion of a certain pair of particles may be distorted. This distortion, in some sense, is a measure of the final-state interaction between that particular pair of particles. It becomes conspicuous especially when the relative velocity of the pair is small, and the other particle, from energy conservation [cf., expression (3b), Part III], can escape with a larger relative velocity without distorting the wave function of the relative motion of the pair. Under these assumptions, the transition rate of the reaction is proportional to the probability that the pair of particles are formed at their range of nuclear interaction, i.e. , at r : : : a apart [Fermi, 1951; Landau, 1965]. As was stated by Landau, this is just the squared modulus of the wave function of the particles formed when they are in the 11 reaction zone 11 multiplied by the size of the phase space region into which the reaction proceeds. The statement can be also visualized by considering the inverse reaction [Watson, 1952]. The probability, that the two final- state-interacting particles be found in the vicinity of each other such that the subsequent transition may occur, is proportional to the squared modulus of the wave function of the pair at r : : : a. The actual form of the wave function at r : : : a is not known, but just for the purpose of estimating the dependence of the wave function on the energy of the relative motion, it is sufficient to 36 continue inward the wave function from the region r > a to r : : : a. As is always implied in the application of the zero-effective-range approximation to the low-energy scattering problems, this is permissible when the energy of the relative motion is small. At low energies, the Schroedinger equation in the region r ;f,, a is essentially energy independent, so the dependence of the wave function on energy in this region is entirely determined by matching its value and derivative to the solution in the external region. Since the pair of particles is created in a state of continuous spectrum moving in a definite direction, t.h e boundary conditions for the wave function should be chosen in such a way that it contains incoming waves and a plane wave in the asymptotic region. The plane wave is replaced by a coulomb wave with only incoming waves in the asymptotic region, if there is also coulomb interaction acting between the pair of particles. This type of solu- tion has bem used by Lane and Thomas (1958) in their treatment of a three-body disintegration in terms of two successive two-body disintegrations. Since only the final-state interactions at low energies are of interest, one may disregard the contributions due to the spin-orbit interaction and the possibility of rearrangement scattering. The wave function in the external region was taken as c = 1 (0 k r c c.R. - U ccs£ ':' I d. d. Y m .R_ ':< (k ) Y m ( ~) c 1 ( 1) 1-m for a particular channel c . )x He re 0 d and Id are respectively the outgoing and incoming wave solutions . the scattering matrix element U The complex conjugate of n was CCS..t taken, since the state described by the expression (1) is just the time-reversal state of 37 the ordinary scattering state. Because of the boundary conditions just described, all of the outgoing waves should be coulomb waves of the form [Messiah, 1962] --z 'IT W 1 = I'(l-i11 )e c c 11 c ik e ~ r c ~ F(i11 ,1,-i(k r +k c c c -+ r)), ( 2) = z 1 z 2 e 2/nv c Comparing the expression (1) with the where Fis the confluent hypergeometric function, and 11 is the coulomb parameter. spherical harmonics expansion of the expression (2), the coefficients xc/ s are found to be x d = 27ri 1-1 e -icr co where crd = arg I'(l + 1 + iric) is the 1-th wave coulomb phase shift. If there is no nuclear interaction, i.e., U '\Ir c is reduced identically to w '· . c 2ir1') e n CCS.x. = exp[2i(crC.x.n - ercO )], The factor c - 1 is known as the coulomb correction (non-relativistically) in many atomic and nuclear problems such as the photoelectric effect and f3-decay where the electron produced in the final state is interacting with the residual nucleus by the coulomb force between them. If one writes w as c 38 '1f =WI+ c c [e -2i(cr - er ) d co ( 3) and evaluates it at r::::: a, one finds that the main contributions come from those terms containing l / r. Since the square bracket in expression (3) :is just the partial-wave elastic-scattering amplitude, the transition rate can be also approximated as something proportional to the scattering cross section of the interacting pair of particles [Watson, 1952; Migdal, 1955]. The approximations discussed so far amount to a factorization of the wave function of the relative motion from a complicated three-body transition matrix element. The possibility of this factorization can also be rediscovered in making .s ome model calculation based on some reaction mechanisms. To the first order of the plane-wave Born approximation, it is found that the transition matrix element, for each channel in the final state, can usually be put in a form as (4) where lfi(r, ~) is the wave function of the relative motion of the pair of particles interacting in the final state and fik is their relative momentum . v(r) is the interaction acting between them and the vector fiq is the momentum transfer to the relative motion of the pair. <j>(r, p), depending parametrically on other momentum transfers denoted by p, is some bound function left over after the integrations over all degrees of freedom other than 1! have been 39 carried out. The expression (4) can be evaluated by assuming a certain form for v(r) , by solving for tµ(1, ~)from the Schroedinger equation and carrying out the integration numerically. Alternatively the integral was simplified by following the idea first used by Werntz Analyzing tµ(t, ~)into partial waves, the expression (4) be- (1962). comes ~ ::! v( r) cj>( r, p) Y 1.. m >:~ . _.,. _,. ,,.. iq · r ..... ( r) e d r, where the radial wave function fikr) is equal to (OJ.. - Us/Ii..) for If only central force is assumed, the angular part of the r >a. integration can be made readily. That is 00 Tfi = ! I/(21..+l)x/Piq • k) Sf;:' (kr)v(r)cj>(r,p)jiqr)rdr , i.. 0 where Pix} is the Legendre polynomial of x of order J... Because v(r) is significant only for r < a and in this region the shape of fikr) is almost energy independent when k is sufficiently small, Tfi is approximated by Tfi = const. x k~ I (Zi..+l)[Iika) - Usi..Oika)]J Piq · k), 1 (Sa) j_ where J 1 = S <j>(r,p)v(r)i/qr)rdr, 0 (Sb) 40 The radial wave function is factored out; each partial-wave amplitude, however, is modified by a factor J £' A cut-off radius can be introduced in the evaluation of the integral (Sb); this has the effect of simulating the distorted waves [Yu and Meyerhof, 1966]. As is noticed from the expression (5a), of symmetry in the angular correlation. q gives an axis Since a particular reaction mechanism is characterized by the momentum transfers q and If, an angular correlation, on the other hand, may indicate information about er and J .R. and tell which one of the reaction mechanisms lS more favorable in describing the reaction. Bo Resonance Parameters of the First Excited State of 4 He. Since none of the interacting pairs of particles was meas- ured in the a-particle energy spectrum of the reaction 7 Li(p, a), one I' has to integrate 1Tfil 2 over all directions of k. From the expressions (5a) and (5b), each partial wave is added up incoherently as (6) ( 7) where is usually referred as the generalized density-of-states function (Phillips et al., l 960]o To include the effect of the n + 3He chan- nel, a real scattering matrix amplitude D s.R. was introduced. In terms of this and the phase shifts, the scattering matrix element U s.R. and thus the expression (7) can be evaluated. As discussed in Appendix A, the scattering matrix element was parametrized 41 in the R-matrix theory by the resonance parameters (a, ER, yp 2 , y z). n The total phase shift os.R. for the partial wave ( s, .R.) is the sum of the coulomb phase shift w.R. = a-.R. - a- 0 , the hard-sphere phase shift and the nuclear phase shift [3 s.R." F.R. and G.R. are respectively the regular and irregular real coulomb functionq. The final form for the expression (7) is given by where Pika) = ka/(F.R.2 + G.R.2 ) is the penetration function. For a one-channel problem D si reduces to unity and p sika) becomes 2 4 sin [3 s.R. Pika) The generalized density-of-states function, expressed in this form, was used before by Barker and Treacy (1962) in analyzing the deuteron energy spectra from the reaction 9Be(p, d) 8 Be,:'. By assuming the contributions from the .R. = 0 and .R. = 2 states of 8 Be only, IJ 0 12 and IJ 2 12 appearing in the expression (6) were left as the free parameters adjusted to fit the experimental energy spectra. 4 Returning to the reaction 7 Li(p, a) He':', the a-particle energy spectra shown in Figure 9 indicated a strong p + 31-r final-state interaction near the higher-energy ends of the spectra. Since a 1 S 42 resonance due to the p + 3H interaction was established from many other sources of experimental data [cf., Part I], it was interesting to see how this resonance affects the a-particle energy spectra based on the expression ( 6). By restricting to the 1S partial wave only, the resonance parameters derived from a least-square fit may be used to compare with existing knowledge about the lowenergy p + 3H interaction deduced from other experiments. = 8. a l = E . be o- . . , while the corresponding value predicted by the Let the experimental differential cross section at 8 and E a J lJ assumed transition matrix element be er ..• By adding together the lJ contributions from the p + 3H and the n + 3He channels, o- .. can be lJ written as .!. l~· D (J •• lJ = f( 8 ) E z nz + 1 - 2D ncos(213 sx. j i sx. n) psx. P /k a) p + D 2 s.R. + 1 - 2Ds.R.cos(213ns.R.)]. P p n(k a) nx. ' n (8) where IJ 0 12 and the other energy independent factors are absorbed into f(8.). l Letters p and n are used to distinguish the p + 3H from the n + 3He channel. The factor l/k in the expression ( 6) is can- celed by an identical one in the phase-space factor given by expression (5), Part III. Since the experimental differential cross section differs from the actual number of counts N .. only by a normalization cons - lJ tant, the root-mean-square error of o- . . was assigned as lJ 43 0- . . Lio- . . lJ =-42 · Nif The fitting procedure involves a search for the minimum of the expression er ij - er ij) ( Lio- .. i.=l j=l (9) lJ where M is the number of energy spectra, and N. is the number 1 = 8.. Table I lists all the a i spectra included for the least-square fit. As was mentioned be- of data points in each spectrum at E; 3 fore, for higher excitation energies in the p + H system, the contributions of the higher partial waves may become important, and at the same time the final-state interactions between other pairs of particles will begin to mask the effect of the p + 3H final-state interaction. Expression (8) therefore is not expected to be valid for higher excitation energies. All the spectra shown in Figure 9 start with a rapid rise just above the p + 3H threshold, reach a max imum at 20. 06 Me V, and then decrease slowly to a minimum near the n + 3He threshold. Since this is a general feature, a cut-off energy of 20. 30 MeV was taken in the x2 evaluations. With this cut-off energy , the number of data poi nts included in the fit reduces to 1ti=l Ni = 229 in a total of 26 spectra . It is possible to apply the pr inciple of the maximum likelihood to determine all of the 30 parameters, i.e. , 26 f(8.)'s plus four 1S 1 44 resonance parameters, by the least-square fit. The resonance parameters (a , ER , y z, y z) are expressed in Appendix A in p n a Breit-Wigner form through D s.R. and f3 s.R.' Table II lists the sets of resonance parameters given in literature with the x . z resulted from the least-square fit. These min sets of resonance parameters differ from each other over a very Unfortunately the existing p + 3H scattering phase wide range. shifts are not accurate enough to pin down those ambiguities. . z, the a-particle energy spectra obtained min in this work give, however, some preference over several sets Judging from the X of the resonance parameters. The reported range of the p + 3 H interaction ranges from 3. 0 F to 4. 2 F. It was fixed in each search for the least square. , y z, y z) we re found R p n by an iteration method discussed in Appendix B .. The results obThe other three resonance parameters (E tained with a = 3. 0 F and a = 4. 0 F are summarized as the following: a (F) ER (MeV) y z (MeV) p y z (MeV} 3.0 20.35 ± 0.01 5.53 ±0.24 2. 88 ± 0. 21 4.0 20. 45 ± o. 02 3.38±0 . 09 2. 23 ± 0. 10 n where based on the Chi-square distribution [cf. , Appendix B], the errors were assigned. eters to the The sensitivity of those resonance param- l is shown in Figure 23, and the p + 3H scattering phase shifts and the differential cross section for the reaction 7 Li(p, a) calculated from those resonance parameters are respec- tively shown in Figures 24 and 25. The curves in Figure 9 are the differential cross sections calculated with the first set (a = 3. 0 F) of the parameters. 45 = 20. 06 Me V for all the spectra are quite x satisfactory, the data points corresponding to this excitation en- The fits at E ergy are taken, without assuming any background contribution, as the angular distribution of the a-particle group leading to the first excited state of 4He. The result is shown in Figures 5 and 26. C. Triton-transfer Reaction Mechanisms in the Reaction 7 Li{p, a). Nothing about how the reaction ends up with a three-body final state was asked in the previous discussions. It was assumed that the particles are produced in the vicinity of one another, and one of them escapes from the recoil pair of particles with high relative velocity. The distortion to the wave function of the rela- tive motion of the recoil pair was due to their final-state interaction only. With these assumptions, a transition matrix element was constructed. From the least-square fit of the calculated differ- ential cross sections to the experimental ones, an angular distribution was taken without assuming any backgrounds. Further justification can be obtained, if the angular distribution itself may be predicted from some model calculation. As is shown in Figure 26, the angular distribution has a forward peak and suggests the possibility of a pick-up process. In this section, various attempts to understand this prominent feature will be discussed. For calculating the transition matrix element in the plane wave Born approximation, the wave functions in the initial and final state were written respectively as 46 w.1 = <1>.dt1 - :Ri>~ [hi1 + 3R1) - it 1J ( 10) ik · [i2 - ..!.. (f 1 + 3R 1 + 3R 2 )] , x e p 7 and wf = X4(t1 - R1)x4'(~2 - Rz)e ik · [·;}(?\ + 3R 1 ) - t(t 2 + 3R 2 )] a ( 11) where the wave function <j> 4 describes the relative motion of a p + 3H two-particle subsystem in the 7 Li nucleus, and <j> 34 de- scribes that of the other triton with respect to the center-ofmass of the two-particle subsystem. x4 and x4 ' are respectively the internal wave functions of the detected a-particle and the reThe vector f. (R . ) designates the position of the 1 J -+ i-th proton (j-th triton), and 'flk ('flk ) is the initial-state proton coil 4He system. ~ . p a (final-state a-particle) momentum in the center-of-mass system. The second term in the expression ( 11) was obtained from the first term by exchanging the protons and tritons simultaneously, and was included because the experiment is not able to tell which proton or t r iton is present in the de t ected a-particle . If the unbound pair interactions in the final state are ne- glected [Banerjee , 1960], the interaction responsible for the triton-transfer reaction contains only V pt (r 2 - R2 ) . matrix element is = (wf, vptw.)1 - Td.i + Tex . The transition 47 Here Tfi is split into a direct and an exchange term, and they are given by the integrals ( l 2a) and x S dt at cj> 34 (~ -iq at I' a )e (12b) t at • _,, Here the new set of integration variables are defined as r ~ pt fers = "l z - R z and :f at = = r 1 - R--+ 1 , -i> t (t1 + 3R 1 ) - Rz, and the momentum trans - 1iqp and 1ic1a are given as qp = ~p + ..!.~ 4 a and ;:}- =.ik~ + k . a 7 p a Y. ,+ ( 13) The corresponding vectors q 1 and q 1 in the expression for T p a ex are obtained from and with k replaced by -'it , p a a a When the recoil 4 He system is in its ground state , i.e ., q q X4 1 = X4• the overlap integral ( 14) 48 appears in both T . and T , and can be taken out as a common d1 ex factor. The ~ -integration may be simplified , if a zero-range pt p + 3H interaction is assumed. That is by setting (15) in the i ntegral [Tobocman, 1961 ]. This approximation was used by Maxson ( 1962) in analyzing the angular distribution of the ground-state ·a-particle group from the same reaction but at some what higher energies, 15. 0 MeV and 18. 6 MeV. The 7 Li nucleus (3/2-) was regarded as a two-particle system consisting of an a-particle ( 0 +) and a triton ( 1/2 +) coupled with orbital angular momentum .R. = l, The wave function corresponding to this model can be written as ( 16) vm where M. is the spin projection of the initial-state 7 Li nucleus. 1 The transition probability, after averaging and summing over the initial- and final-state spin projections, becomes = const. X (w 2 (q ) + w2 (q a 1 a ) + 2w(q )w(q 1 )cos 8), a a ( 1 7) where cos e = q . Cl 1 , and a. a. 00 w( q) = S 3~ x2 U 0 (x)j 1 ( qx)dx. 49 Introducing a cut-off radius R and applying the Butler theory of c surface reaction, w(q) is approximated as [Banerjee, 1960] R w(q) c U (R ) c 34 = - - - - - - [qR c j 0 (qR c ) - (2 + A)j 1 (qR c )]. q z - ~ ( 18) [iZ Here M 34 and E 34 are the reduced mass and the relative energy of the a-particle and triton in the 7 Li nucleus, and A= R [d1nU 34 (r)/drl -R is the logarithmic derivative of U 34 c r- c at R • Estimating A from the known binding energy and substic tuting w(q) into expression ( 17), the calculated angular distribution for 7 Li(p, a 0 ) is shown in Figure 6. As was noticed by Maxson, the angular distribution is very sensitive to the choice of R best value of R and the c for the fit is found to be energy dependent. At c 9. 1-MeV bombarding energy, a cut-off of 5. 3 F gives a better agreement with the angular distribution, but it does not reproduce the slow decrease in differential cross section for the smaller angles. As the recoil 4He system is in its excited state, X•' becomes orthogonal to X•, since they belong to two different eigenstates of the same Hamiltonian. of the 7 Thus if one takes a. + 3H model Li nucleus seriously and assumes that cj> 4 is identical to x4 , i.e., the free a-particle state, the overlapping integral ( 14) is vanishingly small. This means that only the direct term Tdi needs to be considered. With the same approximations, the angu- lar distribution of 7Li(p, a. 1 ) was calculated. It starts with almost a constant value at forward directions and peaks up at backward angles. Since the direct term given by expression (12a) implies 50 that the incident proton knocks out an a-particle and forms an excited state of 4 He with the triton, this process occurs predominantly in the backward directions. In the second attempt , the zero-range approximation implied by the expression ( 15) was dropped . The ground state + and the first excited state of 4He ( 0 ) was considered as in a 1 S state of the two-particle system consisting of a proton and was replaced by the operator a triton. The interaction V pt 'f1Z ( 2M ~ + E pt ) in the 1pt -integration. The radial integral can pt also be simplified by using the same method with which the ex- · pression (18) was derived, as 00 00 + E pt ) r = 2M r pt c S U 4 ( r) j 0 (pr) r 2 d r c U 4 (r )[pr cos (pr ) - (1 +A.)j (pr )]. c c c 0 c Here U4 is the radial wave function of the 4 He system either in its ground state or its first excited state, and A again is the logarithmic derivative of U 4 at a cut-off radius of r . The logarithmic derivac tive was determined by the binding energy for the ground state , while, for the first excited state, it was evaluated by the asymptotic = 20. 06 MeV) x A peak in forward angles of the angular dis- expression (1) incorporated with the phase shift (at E obtained in Section B. 7 tributions of both Li(p, a. 0 ) and 7 Li(p, a.i) can be generated, but for the latter it was too small and too broad in comparison with the data [cf., Figure 26]. 51 It was therefore thought that the exchange term should be taken into consideration in order to reproduce the shape and magnitude of this forward peak. It is possible that the p + 3 H two-particle subsystem in 7 Li may have a definite probability of being excited to the first excited state of 4He . For convenience in computing the relative amplitude of the direct and exchange processes, the wave functions involved in the overlap integral ( 14) were assumed as and Here f 0 (kr) is the radial wave function of the first excited state of 4He, and for r >a, it becomes the 1S component of the expresThe inverse-square decay length f3 2 of the sion (1) in Section A. p + 3H subsystem in 7 Li is not known, and is arbitrarily taken as that of a free a-particle, i.e., '( 2 = 0. 21 F - 2 • The wave function U 3.dr) in expression (16) was chosen as a Gaussian type given by where a. = 0. 28 F- 1 is chosen to fit the reduced width of the ground state of 7Li [Tombrello and Parker, 1963]. Since f 0 (kr) is unknown for r J;;; a, the integral involving x4 ' cannot be carried out analytically. They were approximated [cf., expressions (Sa) and (5b), Section A] as 52 and -- pz +E pt * ) x 0 f 0 ':' (ka) 4n 2nk p2 -·- The factor fo '''(ka) = Io(ka)-U 00 0 0 (ka) appears in both Td. and T , I ex and it can be factored out in the total transition matrix element. The differential cross section therefore contains also the same generalized density-of-states function as the expression (6) does. As in the previous approximation, a cut-off radius of 5. 2 F was needed for the integral 00 00 S 2 U3.dr)r jiqr)dr:::: 0 SU 3~(r)? R j/qr)dr c and the Simpson's rules were then used in obtaining the integral in the right hand side . Figure 26 shows the results of this calcula- ti on. A satisfactory fit, with those simple calculations, to the experimental data seems to be not very likely. It however does indicate qualitatively that the angular distributions of both 7 Li(p , a. 0 ) 53 and 7 Li(p, a. 1 ) may be understood in terms of the triton-transfer mechanisms. The assumptions made in Section B for deducing t he resonance parameters are plausible. D. A Modified Born Approximation Calculation for the Reaction D( 3He,p) . The angular correlations from the reaction D( 3He, p) were obtained at the proton energies chosen such that the excitation energies of the p + 3H system lay between 20. 66 MeV to 22. 06 Me V. As was mentioned in Part I, the P-wave interactions are important in this energy 1966]. range [Meyerhof, 1965; Yu and Meyerhof, The observed anisotropy in the angular correlations pro- vides additional evidence about this. The modified Born approxi- mation calculation , to be discussed in this section, was made to estimate the relative amplitude of producing the p + 3H and n + 3He final-state interacting pairs in S-wave and P-wa,ve states. In terms of the usual analysis of the particle-particle angular correlation, this is equivalent to an estimation of the density matrix of the 11 intermediate '4oHe 11 formed during the reaction. In Yu and Meyerhof 1 s analyses of the proton energy spectra from the reaction D( 3He, p) and the neutron energy spectra from its mirror reaction 3H(d , n), the following reacti on mechanisms were considered: A (D) 3 He picks up a neutron from the target deuteron and forms a p + 3H (n + 3He) interacting pair. B (E) 3 He strips its deuteron to the target deuteron to form a p + 3H (n + 3He) interacting pair. 54 3 C (F) He breaks up g1vmg one of its neutron (proton) t o the deuteron, the 3H (3He) formed then interacts with the other proton (neutron) from the breakup. 3 G He interacts with the neutron from the direct breakup of the target deuteron. In a phase space region where the nucleon +nucleon final-state interaction is of less importance than those of p + 3H and n + 3He , the deviation of the final-state wave function from pure plane waves can be attributed to the effects of the p + 3H and n + 3He final-state interactions only. By using the method described in Section A and the Wigner type of the nucleon-nucleon potential, Yu and Meyerhof have given the transition matrix element for any of those mechanisms. as 2 -k . /4A. 2 M . = n. I. e 1 1 1 1 1 I iP_eCpi · p"). (21 + 1) Nip") J.R. ' .R. A Here i stands for each of the m.echanisms listed above, and p . and 1 are respectively the unit vectors of the momentum transfer to the relative motion and of the relative momentum of the final-state interacting pair. Other quantities appearing in this expression will be r edefined in the following dis cuss ions. The a ngle e . b e tween p. 1 1 and the beam axis will det e rmine an axis of symmetry i n the angular corre la ti on , if the reaction goes predominantly by the mechanism i. For the present experiment with E calculated as p = 7. 8 Me V and e p = 30°, e.1 is 55 Process i e . (deg.} A (D} 16.3 B(E} -16.8 C (F) - 4.0 G -41. 0 l The observed angular correlations have an axis of symmetry along the •He recoil direction, which was found also to be lying at -16. 8° with respect to the beam. 3 The mechanism B (E), representing the He stripping of its deuteron, is therefore assumed to be the dominant process in the analysis . If one labels the nucleons in the target deuteron by 1 and 2, and those in the incident 3He by 3, 4 and 5, the initial- and finalstate wave functions can be written as and x e (11~ ) is the initial-state 3He (final-state proton) momenp tum in the center-of-mass system. x~ I ' as was defined in Section where i'lk T C , describes the relative motion of the final-state interacting pair of a nucleon and a mass three nucleus. x, cj>, and Lj; respectively are the fully antisymmetrized internal wave functions of mass one, mass two and mass three nucleus. They are assumed to be separable in 56 space and spin-is ospin variables, and are given by ( 1) One-nucleon system rn1 X(l) = P1 11 1 f.l.1 where p, 71 are the spin and isospin mat rices. (2) Two-nucleon system 3 S 1 state with Md,O D \ (l,2)=f._.; P1 x (3) m1 Pz mz TJ1 f.l.1 'Y'Jz f.l.z Three-nucleon system in 2S 1 state [Schiff, 1964] . 2 l\i(l1, it'z, t 3} = Rnv ( 1, 2, 3) ("13 /rr) 3/ 2 y 3 2 x e -y [ t 111 -lz1 + lt3 - }(t1 +~z )1 J, 2 and R nv (l,2,3) ="161 x P3 mz 111 f.l.1 1lz f.l.z \ L TJ3 f.l.3} • 2 57 In accordance with Yu and Meyerhof, the inverse decay lengths of the bound wave functions were taken as a= 0. 167 F- 1 and y = 0. 36 F- 1 respectively for the two-nucleon and three-nucleon systems. V = V 13 The interactions responsible for the stripping of 3.He are + V H: + V Z3 + Vu, where Vij was generalized to include the spin and isospin dependences as a scalar type with Gaussian shape given by v .. = -V0 e -f3zl~ -~ 1z i j lJ ..J... (W , BP .. lJ (J' CT MP .. P .. lJ lJ T T HP .. ), lJ (19) CJ' with f3 = 0. 63 F- 1 • T The exchange operator P .. (P .. ) acts on the lJ lJ spin (isospin) variables of the nucleons i and j. The two sets of constants W, B, M and H used are [Preston, 1962] type w B M H Rosenfeld -0. 13 0.46 0.93 -0.26 Serber 0.5 0 0.5 0 Because both \II. and wf are antisymmetric under the ex1 change of nucleons 1 and 2, V lj and V j have the identical matrix 2 elements. The transition matrix element for the process B(E) is then reduced to By expressing the spin wave functions of the final-state interacting pair in its channel spin representation, i.e., 58 Tf. was separated into a space integral and a spin-isospin over1 lap as Tfi = const. x I \ L (z-z-iim.f:SM) L \' i i SM < ii , µ:. ms r1s P.s , I/S,jl.f:ii) SSM (1,2,3;4,TJ.f: Ps j=3,4 (20) Md, O nv o1 j D ( 1 , 2) R ( 3 , 4 , 5)) • A bar was put on the final-state spin or isospin projection to distinguish the corresponding quantity in the initial state. The operator o j is an abbreviation for the sum of operators with i = 1 1 in the parentheses of the expression (19). Its matrix element 25 1 + (o .) and the space integral I., for j = 3 or 4, are calculated 1J J in Appendix C . After averaging over the initial spin projections Md and n, and summing over the final state spin projections fi, m4: and ms, the squared modulus of Tf· 1. becomes !Tf/~ = const. x l s Tr ( l \(S,p..v) i=3,4 25 1 + (o 1 j))( I I/Si!•v )ZS+l(0 j=3,4 The cross sign indicates that both the transpose and complex conjugate of the matrix were taken. fined as In terms of the abbreviations de- 1 /}t 59 and 2S+l C = -I.dS,p.4ii)M. The final form for !Tfi 12 is given as the following: ( 1) For D + 3He _.. p + 3He + n, i.e. , p. 4 - =-i" and 1 v = +z ( 21) 3.-. 3C + R e ( i2s . 3A3B>:< + },. -n (2) For D + 3 He _.. p + 3 H + p, i. ~. J P.4 = +z1 an d -v =•z1 ( 22) K here is just a numerical constant independent of which one of the branches the reaction leads to . As is indicated in Appendix C, the p + 3H and n + 3 He finalstate interactions are taken into account in terms of the scattering matrix amplitude D si. and the respective phase shifts 8 ps.R. and o ns.R. by the factored-wave-function method discussed in Section A. the 4 Heexcitationenergies 20.662:SE x :S22.055MeV, D n• sx. o For n psx. and ons.R. given by Bransden et al. (1956) and by Meyerhof and McElearney ( 1965) are listed in Table III. Using an a = 3. 0 F and 60 a cut-off radius R c = 5. 0 F [Yu and Meyerhof, 1966], the angular = 7. 8 MeV with Rosenfeld type p nucleon-nucleon potential were compared with the experimental correlations calculated for E correlations in Figure 27 by a least-square fit. The dashed curves represent the calculations from all the phase shifts and scattering matrix amplitudes given by Meyerhof and McElearney; while the solid curves are the same c.alculations except the triplet S-wave n + 3He phase shift was changed to the value of Bransden et al. For partial waves other than the triplet S-wave , the n + 3He phase shifts from Meyerhof and from Bransden agree in signs. Although Bransden 1 s values are smaller, the data are not able to indicate any definite preference between them. It is clear that the set of the n + 3He phase shifts from Meyerhof et al. cannot reproduce the observed branching ratio and the forward-to- backward peak ratio in the p - 3 He correlations. In the phase-shift analysis of Meyerhof and McElearney, a discrepancy was found in predicting the energy dependence of the ratio B /B , where B /k z nn 1 nnO nnL n is the coefficient of PL (cos en) in the Legendre polynomials analysis of the n + 3He elastic differential cross section. e · n and k n are the direction and magnitude of the neutron wave vector in the centerof-mass system. This discrepancy was considered to be caused by an incorrect n + 3He phase shift. The n + 3He phase shifts of Bransden et al. from the resonating group calculations, whose T = 1 phase shifts were found to be in good agreement with the p + 3He and n + 3 H elastic-scattering data [Tombrello, 1965 and 1966], were used in computing the curves shown in Figure 21. 61 The normalization constant obtained from the least-square fit to the p - 3 H and p - 3 He· angular cor r elations at 7 . 8 MeV was used to generate those correlations taken at other prot on energies. It was found [cf., Appendix C] that the cut-off radius R sensitive to the quality of the fit, and 14 is not very c is very much smaller than 13 for R chosen to be somewhat greater than 2 . 0 F. This means c that only 13 contributes to the differential cross section, and the constants that describe the nucleon-nucleon potential appear effectively in (W + M - iB - tH) 2 as a constant of proportionality. Consequently, the present data are not able to tell the difference in choosing among the nucleon-nucleon potentials. The agreement of the calculation with the angular correlations becomes poor for lower proton energies, i.e., for higher 4He excitation energies, this may be partially due to the inappropriate phase shifts and also due to the fact that the factored-wave-function method may become less efficient. 62 V. DISCUSSION OF RESULTS Nuclear reactions with three outgoing particles are very complicated in comparison with the reactions wher e there are only two particles in both the initial and final states. Recently substan- tantial progress has been made in the study of three-body problems by Faddeev (1963) and Lovelace (1964.). For each of the rearrange- ment scatterings, the transition matrix is solved by a system of inhomogeneous integral equations. The inhomogeneities and kernels of these Faddeev's equations are directly related to the two-body transition matrices. Besides its involved mathematical technique, the method is still too difficult for an actual application at the pre sent stage. This is because the kernels also require the two-body transition matrix element off its energy-shell. More experimental data over wider ranges therefore are needed to obtain such information by analytic continuation. The factored-wave-function method or the Watson-Migdal . approximation (referred as WMA) used in the analyses of Part IV, is crude in the sense that it is just a very primitive approximation of the solution to the Faddeev 1 s equations [Gillespie, 1964]. From the results obtained in this work, the approximation however appears to be better than expected. An explanation may be tha t for the phase space region of interest, the final-state interaction of one pair of particles dominates over that of the other pairs in the reaction. All the a.- particle energy spectra of the reaction 7 Li(p, a.) at 9. I-Me V bombarding energy, and 2. 5° :5 e :5 120°, showed an a. enhancement of the differential cross section over the phase-space 63 factor near the highest possible energies. The WMA was used to predict the spectral shape for the higher energy parts of these It turns out that these parts of the spectra can be suffi- spectra. ciently understood by assuming a resonance in the 1S p + 3H system. The 1S phase shifts derived from a least-square fit are shown in Figure 24 to compare with those given in the literature. Both the open and dark circles with the error bars represent the solutions of a point-by-point phase shift analysis of the p + 3H elastic scattering data by Kurepin et al. (1966). The triangles and the squares indicate some of the phase shifts obtained by Werntz (1964) and Meyerhof and McElearney (1965) respectively. In addition to the p + 3H elastic scattering data due to Jarmie et al. (1963), the total cross section of the 3H(p,n), 3 He(n,p) and 3He(n,n) reactions [Seagrave et al., 1960] were also included in the latter analyses. From the discontinuity of the energy derivative of the p + 3H elastic scattering differential cross section, Werntz was able to argue that the 1S p + 3H phase shift at the n + 3He threshold energy should be equal to 108°. With the resonance parameters obtained in this work, the phase shift at this energy was found to be l 04 ° for a = 3. 0 F or 101 ° for a = 4. 0 F. The agreement of the phase shifts with those obtained in this work from a rather indirect approach is gratifying. As was explained in Section B, Part IV and in Appendix A, the phase shifts here were found parametrically through a set of resonance parameters (a, ER, y z, y z ). Although in the leastp n square fit, the x 2 for a given range of nuclear interaction, a, appears to be quite sensitive to the rest of the resonance parameters 64 near the solutions [cf. , Figure 23], the re still is no definite as su rance that the set of parameters (ER, y 2 , y p given a is unique. (0.025 eV) on a ER 3 2 n ) obtained for a The total cross sections by therm.al neutrons He are calculated as the following: y 2 p y 2 n Wigner limit (b) nn calc. expt. np calc. (J 0- (b) expt. 3.0 20.35 5.53 2.88 9.2 1. 90 1. 80 12400 5280 4.0 20.45 3.38 2.23 5.2 2,70· 1. 80 15000 5280 Both sets of the resonance parameters predict approximately the magnitudes of the thermal cross sections given by Seagrave et al. Besides the phase-space factor, the WMA [cf., Section B, Part IV] predicts an angle-independent a-particle energy spectrum. An additional justification for the use of this approximation was obtained by making a model calculation to predict relative yields in the a-particle energy spectra taken at various angles. As was shown in Se ct ion C, Part IV, the angular distribution of the a-particle group leading to the first excited state of the 4 He system can be qualitatively understood by a PWBA calculation of the triton-transfer reaction mechanisms. In order to explain the forward-peaking in the distribution, it was found that the exchange process has to be included in the calculation. It may be recalled that a cut-off in the proton center-of- mass energy of O. 485 MeV (E x = 20. 30 MeV in 4He system) was used during the search for the least square in the fit of the spectral shapes to the a-particle energy spectra. two sources of complications come in. As the energy gets higher, The first is that the contri- 65 butions due to the final-state interactions of the higher partial waves have to be included. The second depends on the type of particle 3 and its relative velocity with respect to any of the other particles in the final state of the reaction . As the exci- tation energy of the recoil pair becomes higher, one may reach a phase -space region in which particle 3 is interacting strongly with one of the other particles. The angular-correlation meas7 urements from the reaction Li(p, a.) [cf., Section B6, Part II], indicate that the effects of the p + 3H and n + 3He final-state interactions are masked by the strong final-state interactions between the a. + H, a. + 3H and possibly a + N pairs, when the 4 He excitation energy is changed from 20. 01 MeV to 21. 27 MeV. The a. - 3 H plus a. - H angular correlations at 20. 01 MeV, inci- dentally, showed that only the p + 3H final-state interaction is important at this energy , and gave an additional confirmation about the 0 + spin-parity assignment of the first excited state of the 4 He systern. To reduce the effects due the second complica- tion just described, the reaction D(3He, p) [cf., Section C , Part II] was investigated. With 16. 5-MeV 3He bombarding energy, the angular-correlation measurements were carried out by detecting the protons at 30° and at 6. 6 MeV $ E $ 8 . 6 MeV. The p excitation energies between these protons and the mass -three particles are higher than that of the 4 He system recoiled. Their corresponding final-state interactions are expected to be weaker. Although the excitation energy of the p + H or p + N system may be, in very few occasions, small, the effects of the p + H or p + N final-state interactions were not seen in the angular correlations. 66 The p - 3 H, p - H and p - 3He angular correlations have an axis of symmetry along the •He recoil direction. From this and the calculations made by Yu and Meyerhof, one was able to conclude that the reaction proceeds predominantly by the 3He stripping mechanism [cf., Section D, Part IV]. Based on this mechanism, a modified Born approximation with a more generalized nucleon-nucleon potential was carried out. The modi- fication was made to account for the p + 3H and n + 3He finalstate interactions by the factored-wave-function method. By using the scattering matrix amplitudes and the p + 3H phase shifts reported by Meyerhof and McElearney to describe the corresponding final-state interaction, the calculation reproduces both the energy and angle dependences of the p correlations. 3 H (Because the protons and tritons from the decays of the recoil 4He systems were emitted oppositely in their centerof-rnas s system, the p - H correlations we re converted into the p - 3H c .o rrelations.) It was thought, therefore, that the as sumptions regarding the 3He stripping and the factored-wave-function method are plausible. the p - 3 The analyses were then carried over for He correlations. In order to predict the p - 3H to p - 3He branching ratio and the forward-to-backward-peak ratio in the p - 3He angular correlation , it was shown that at least for the triplet S-wave the n + 3He phase shifts due to Bransden et al. are preferred. As was remarked also in Section D , Part IV , the agreement of the calculations with the angular correlations becomes 67 poor for lower proton energies (in particular. for the p relations). 3 He cor- Since the relative energy of the n + 3 He system is 0. 764 MeV lower than that of the p + 3H system, one would be rather reluctant to assume that the factored-wave-function is not valid. It was stated in Meyerhof and McElearney 1 s analyses that the region of validity of the scattering-matrix amplitudes and the phase shifts they obtained does not extend to the region beyond approximately 21. 3 MeV. The phase shifts due to Bransden et al. from the resonating-group calculations were derived without considering the inelasticity, and are expected to be invalid for higher relative energies. It is clear that one therefore needs to obtain a better set of the scattering-matrix amplitudes and the phase shifts in order to say whether the theory of the fina-1-state interaction employed in this work is still applicable for the higher relative energies of the pair of particles interacting in the final state. 68 APPENDIX A. PARAMETRIZATION OF THE PHASE SHIFTS For -!He excitation energies well below the d + d threshold energy (23.841 MeV), there are p + 3H and n + 3He channels to be considered. The scattering matrix satisfying the symmetric and unitary condition can be written as a 2 x 2 matrix [Meyerhof and McElearney, 1965] for each partial wave of channel spin s and orbital angular momentum 1, u si. u pps£ u pnsi. u npsi. u nns.P. = 2i6 D si. e n +0 i( 0 l psx. ± i( 1- D 2) 2 e psi. ) ns.P. s.P. 1 D 2 )z ... s£ e .(1 ±i ( 1) i( 6 psi. +6 nsi. ) 2i6 Ds£ e ns i. n(o J is the total phase shift of the p + 3H (n + 3He) system. psx. nsx. The real quantity Ds£ is the scattering-matrix amplitude which couples where 6 the two channels together. If there is a resonance in the partial wave . ( s, 1), the phase shifts and the scattering-matrix amplitude can be parametrized by a set of resonance parameters in the Breit-Wigner forms. The d + d channel, belonging to a negative-energy channel, is eliminated by using the method due to Teichmann and Wigner [Lane and Thomas, 1958]. 69 Following the notations and definitions of Lane and Thomas, the scattering matrix is connected to the nuclear parameters through the r educed R-matrix for E I [(1-R upp = p 0 L nn n )(1-R n > 0 as the following: 0 L o>:') - R R L L 0 ':'] pp p n p pn n p 0 0 0 0[(1-R L )(1-R L )-R R L L p nn n pp p n p pn n p U np = 2iP R p np /[ ( 1 -R L nn n 0 )( 1 - R L pp p 0 ) - ( 2a) 0 R ] 0 R L L np pn n p 0 ), (2b) and sim.ilarly for U and U with the subscripts p and n inter. nn pn changed. Various quantities here are defined as a = the range of nuclear interaction c M E c c = the reduced mass of the particle-pair in channel c = the energy of the relative motion of the pair 1 k = (2 M c E /11 2 c c f is the relative wave number of the pair = kc a c I = the incoming wave solution in channel c c 0 c = the outgoing wave solution in channel c sc = the shift function p = the penetration function B I c c 0 c = the boundary value i n channel c = S c +iP c B . c 70 In general the reduced R-matrix elements are expressed in terms of the reduced-width amplitude R where cc 1 \:>._ of the resonance labelled by A. as =L \ (A-I)A.A.' =(EA. - Ep)<'\x. • - L (3) "ex. 'leA.'(Se-Be)' e and E is chosen as a common ene rgy reference. The channel inp dex e runs through all the negative-energy channels. For En$ 0 , i .e., Ep =En +Eth$ Eth' where Eth is the threshold energy of the n + 3 He channel, the summation in expres- sion (3) now has to include then + 3He channel, also. The corres- ponding scattering matrix and the reduced R-matrix, written as . U and R, are related to each other by upp I = _R. 0 (1-R p (1-R L O>!<) pp p L pp p (4) 0 ) and the other matrix elements U , U and U are no longer of pn nn np physical significance. If a single level A. = (s, .R.) is effective in determining the main feature of the scattering process in the energy range of interest, the reduced R-matrix elements are given by expression (3) as 71 R and for E R p = pp = np \~s.f y ps.f (Sa) {Es.f-Ep) +Ads.£ yns.fyps.f (Sb) (E s.f-E p) + ~ds.f > E h' and as t = _ _ _Y....._p_s_i_Yp.__s.f_ _ _ __ R pp (E n-E sx p + Ad ) n + A n sx_ nsx ( 6) ' for E ::S E h . R and R can be obtained in a similar way or p t nn pn directly from the expressions (Sa) and (Sb) by e x changing the nz (S n-B ) is the ex CA'. Substituting (Sa) and (Sb) into (2a) a;nd {Zb), and subscripts p and n . level shift. The quantity A n= - y CSA'. CSA'. (6) into (4), the scattering matrix elements become upps.f = Es.£ - .R. - I' n) psx e i E s.f E n - sx. 2i(w .R. + cj> .R.) p p , (7a) - 2 (rns.R. + ~s.R.) r unpsi. = zi (I'ns 1 ns.R. zr .!. z ps.R. 2i (I' n t I' ) nsx ps.i:. e i(w +w +<j> +cj> ) n.f p.R. n.R. p.R. (7b) and similar expressions for U n and U for E > E h ' and nnsx. pnsx.0 p t 72 E upps.2 = E 1 s.R. s.R. +- r 2 ps.R. i -- r psi. 2 2i(w e p.R. + <j> ) p.R. ' (8) 2 s Eh. Here r n = 2y n p n is the level width, and Esn p t CS.I'. CS.I'. Cx .I'. is an abbreviation of the quantity E n - E + ~d n + ~ n + ~ n sx p sx nsx psx Finally, by comparing the expressions (7a), (7b) and (8) for E with the expression (1), the phase shifts and the scattering-matrix amplitude are given by {l) E p !3 >E th n = tan - l {I' psx A r- nsi. nE n/ [E psx = tan-l {I' S,r. n2 sx + t (I' _ si- ( 2) E p s E [i 1 I' n2 ) ) } psx E / [E s.R. 2· - t(I' ns.R. 2 - I'psnx 2 )]} nsi. si. Ji and D n2 - nsx r ns.R. r psi - Es/ +t(I'ns.R.+rps.R.)2 th f3 psxn = tan- 1 (I'psxn/2E sxn) and D si = 1. The total phase shift o n is CS.I'. the sum of the nuclear phase shift · !3 cs.R.' the coulomb phase shift w d and the hard-sphere phase shift 73 The level shift D.dsi.. can be set to be very small across the resonance energy Esi. by choosing Bd = Sd/Esi..). The other boundary values B n and B n are fixed respectively by the conditions that o n(E n) PL nL psx sx = rr/2 and D. nsxn(E sxn) = 0. In actual calculations, the shift function (a = 3. 0 F) ford + d channel varies by 13%, and that for p + 3H c channel varies by 240%, per MeV in the energy range of interest. Furthermore because the deuteron reduced width was estimated [Werntz and Meyerhof, 1968] to be about one. half of the nucleon reduced width, 6 dsi.. was set identical to zero in the calculations of the spectral shape for the a-particle energy spectra [cf., Section B, Part IV]. The expressions for the phase shifts and the scattering- matrix amplitude are then identical.to those given in the work of Meyerhof and McElearney (1965). Including the 1S resonance of 4.He only, i.e., X. = (s,i..) =(0, O), the nuclear phase shifts and the scattering matrix. amplitude, f3 n 00 , calculated from an assumed set of (a, E , y 2 , y 2 ) 00 R p n 2 2 , yp00 , ynOO ) were used to evaluate the expression (8) in and D =(a, E 00 Section B, Part IV. a f3 pOO' The procedure for the least-square fit for = 3. 0 F or 4. 0 F is explained in Appendix B. 74 APPENDIX B. A SEARCH FOR THE LEAST SQUARE BY ITERATIONS In estimating a number of parameters a = {a 1 , a 2 , • • • , a s } in a theory from N ~ s measured quantities x 1 , Xz, • • · , xN by a least-square fit, a search is made for the minimum of the quantity ( 1) where <r. is the standard error in measuring x., and£., containing l . l l the set of parameters a, is the theoretically predicted expression for x.. If a = {a1, az , ... 'a } are the values of the parameters a s of the maximum likelihood, then l ~ 8vz) ~ oa m = - a=a F m (a)= 0, with m running from 1 through s. ( 2) The problem therefore is es- sentially to solve for a from this system of equations. If all £. 1 s in the expression (1) are linear functions of a, l the system of equations (2) becomes an inhomogeneous linear one, and can be solved readily by the standard matrix algebra [Mathews and Walker, 1964]. On the other hand, ifs. 's depend on a in a com1 plicated fashion and a linearization of the system of equations (2) becomes too involved, the iteration method [Janossy, 1965] to be described here sometimes proves to be very useful. 75 If a 1 are It starts with an approximate set of values a 1 • chosen to be reasonably close to a, F power series in a - a 1 around a 1 • m (a) can be expanded in a One then obtains in this way an approximate relation between a and a I given by F m (a') = \ ' /, ~ ~ Foam(a))\ n (a' - a) a=a' (3) n. a solved from this system of equations are still the approximate solutions , unless a 1 are very well chosen to start with. By sub- stituting a for a' in equations (3), another solution of a can be obtained, and should be closer to the real solutions step by step in each iteration. The matrix containing (3 F symmetric s x s matrix . (a)/8a ) is a m n a=a 1 The x2 has to be evaluated at (2s + 1) + i s(s - 1) points around a' in the s-dimensional parameter space in order to construct this matrix, The parameters involved in deducing the low-energy properties of the p + 3H system from the a-particle energy spectra [cf., Section B, Part IV] are (ER, 'I 2 , 'I 2 ) and the f(8.)'s. The latter p n i 2 2 were found for each input of (ER' 'I , 'I ) by minimizing the x2 p n se.parately for each spectrum. Since the ~. 1 s are linear functions l 1 of the f(8.) s , these were solved by the first method . l The x2 was then considered as a function of (ER, 'I 2 , 'I 2 ) , and was minimized p n by successive iterations. Depending on the first approximate values, usually four or five iterations were sufficient to bring the parameters in the neighborhood of their optimum values. The final values of the parameters were varied around to check if x2 was really a minimum there . 76 The expected magnitude of X z. from the expression (1) min is equal to N - s, i.e., the number of the degrees of freedom. In the actual fitting x 2 . , however , was found to be considerably min larger than N - s. This may be due to the fact that the standard errors were underestimated and that the theoretical expression is not equally valid for all the data points. The x2 obtained by re- normalizing the IJ'. z, i.e. , increased by a factor of x 2. /N - s, i min should now follow the Chi-square distribution. From the tabulated values for the Chi-square distribution, it was found that the probability that x2 /(N - s) exceeding unity is 50% for N - s > 30. If two of the parameters are fixed, the probability that x2 /(N - s + 2) not exceeding unity is also 50%. The error in determining each of the parameters (ER, y z, y z) was taken as the range of that p n parameter for which the x2 increases by two when the other two parameters were held at some values in the neighborhood of the optimum ones. 77 APPENDIX C. THE SPIN-ISOSPIN OVERLAPS AND THE SPACE INTEGRALS IN THE MODIFIED BORN APPROXIMATION CALCULATION FOR THE REACTION D(3He, p) . . . . over 1aps ZS+ i (0 lj ) , Th e eva 1uahons o f t h e sp1n-1sosp1n defined in Section D, Part IV for j = 3 or 4, involve systematic bookkeeping of the Clebsch-Gordan coefficients and the Kronecker deltas. The latter resulted directly from the orthogonal properties of the spin and isospin matrices. the matrix Writing the operator explicitly, <o .) is given by . 1J (0 J.) = W(l) 1 CJ T CJ T + B(P .. ) - M(P .. P .. ) - H(P .. ) • lJ lJ lJ (1) lJ The operator P .CJ(P .T) exchanges the spin (isospin) projections 1J . 1J of the nucleons 1 and j in the initial-state wave function. Because of the symmetry property of the wave functions used for the two- and three-nucleon systems, it was found that ( 0 lj) can be expressed as some linear combination of the three matrices ( 1), ( P (P 14 CJP 14 T). 14 o;) and In terms of a 2 x 2 matrix, defined as where Md(M) is the spin projection of the initial-state deuteron (the final-state ~e system), the matrix elements of (1), (P p <_ 14 o-p 14 T) are given . by 14 CJ) and 78 -1 (1) = - - 6 b"12 1 l l 0 0 - -6 (22P..vlOO)[o_ (X -X ) ffi5,n S,l M,Md ms,n -n,-n n,n + 26 er <p 14 ) = -1 4"12 0 0 ms, -n l:J. 12 0 ms, n S, 1 M, Md x(o_ ms, n (X x -n, n ], S+l 11 (zzf.J.•v - -100) ~x )+2o_ X ] -n, -n · n, n ms. -n -n, n and x (X -n, -n +X ) ]. n, n . The index n(rns) is the spin projection of the initial-state 3 He (finalstate proton). Depending on the channel spin S of the final.,. state interacting pair, i.e., of the final-state •He system, these matrices are the 6x2 (2S+l) matrices given as the following: (1) (1) For producing a singlet final-state interacting pair =+ l 6"12 0 0 0 0 -~ 0 . -"12 0 -1 0 -1 0 0 l 0 1 0 "12 0 .J2 0 0 0 0 1 (Pl4 ) =+ 1202 (J' 79 and (2) For producing a triplet final-state interacting pair -1 (1) = - - u + 1 6"12 6"12 T Cf and er 1 -1 (P 14 ) = - - u i : 4"12 12"12 v, (Pl4 pl4 ) v -1 --( 1 + l )U 6"12 where U is a 6 x 6 unity m.atrix, and V is a 6 x 6 matrix given by v = -1 0 0 0 0 0 0 1 0 0 0 0 _,.rz - in. 0 0 0 0 0 0 0 0 ""12 0 0 0 0 _,.rz 1 0 0 0 0 0 0 -1 The upper (lower) sign in those expressions describes the reaction 3 He + D --;. p + p + 3H ( p + n + 3He). The other rn.atrices are simply J (Pl3) = - t (1) , (Pl3) =z-(l), (Pl3 pl3) =-(1) T 1 Cf T By substituting those expressions into the expression (1), the trace of (0 .)( 0 .)t, for i, j = 3 or 4, can be found readily, and the re11 1J sults were summarized in the expressions (21) and (22) of Section D, Part IV. 80 To carry out the space integral I. that also appeared in J Section D, Part IV , one needs to choose an appropriate set of integration variables. The purpose is to find a coordinate system in which the integrations are separable. In the center-of-mass system, there are four degrees of freedom to be integrated, since the sum of the coordinate vectors of the five nucleons vanishes. With the bound wave functions assumed for the deuteron and the mass three systems, and with the nu.cleon-nucleon potential assumed, I. is given by J I. = 5 J x S -i~ [i\-i(r1+tz+r3+r.d] -'/[ilt1-l'zl 2 + lt3-f(r1+rz)lz] e p e x4 *[rt4 -i (l 1 +rz +r 3)]e 1 - f3 z Ir 1 - :tj Iz e ( 2) Transforming to the new integration variables, defined as and the integral (2) becomes (3) 81 Here the jacobian of the coordinate transformation is 1/5, and the vector V. is J v.J =a.~J +byJ z where depending on whether j = 3 or 4, a 3 (a 4 ) and b 3 (b 4 ) are reThe integrations over r spectively equal to 0 (-1) and -1 (-1/3). and y1 can be done easily by using the formulas 3 S e 2 -c x 2 2 2 ip ·st dx = (~-; e -- ) e -p /4c c (4) The integral ( 2) is now reduced to -iyzl-fyz-plz - la.p+b.y 1213z(4a.z+3yz)/(4a.2+3yz+13z) x e J J 2 dpdyz. Finally to separate the last two integrations, an additional change of the coordinate system was made. 2!1 = p If one writes and and chooses f. such that the coefficient of t 1 • ~z in the exponent J vanishes, the integration over t 2 can be carried out by using the formula (4) again. 82 to be · The final result is then given by Ij = 81T 3[ z z z z ]- 3/Z y (4n +3y +13 ) szbj-+--,-f-y=-z ] 1T [ 3/Z e -I-tit T -Rp IZ/4yZ ( 6) t.=(}f.-l)(i-~ -ik) where and J J p T £. z = J By a method similar to that used in the general discussions of Section A, Part IV, the wave function x~'(z 1 ) is then expanded into partial waves, and the angular part of the i ntegration can be done readily. The last integral in expression (6) becomes Here fikz 1 ) is the .£-th wave radial wave function of the final-state interacting pair , and the corresponding partial-wave amplitude x 1 83 was given in Section A, Part IV. Since f/kr) is not known for r;;;., a , the factored-wave-function method is used to approximate the radial integral as -s.z rz j/p/)e A cut-off radius R in the reaction. is introduced to simulate the distortion effects c The. integral, 00 J. J, p_ - s -s .zrz j /P/)e R J rdr, c gives the relative amplitude of producing the final-state interacting pair in the P.-th wave by the interaction between nucleons 1 and j. Together with the x 2 resulted from the least-square fit to the experimental angular correlations J . n1 s J, .l: are listed in the following for the cut-off radius varied from zero to 6 F. ·R J 2 c 0.0 4.70 1. 41 1. 06 0.06 992 548 1. 0 4.22 1. 41 0.66 0.06 869 501 2.0 3. 07 . l. 22 0. 16 0.02 825 485 3.0 l. 79 0.89 0.02 3xl0- 3 778 417 4.0 0.84 0.52 6xl0-~ Ix I o- 4 731 349 5.0 0.3 1 0.25 8 x 1 o- 6 2 xi o- 6 707 283 6.0 0.09 0.09 4 xi o- 8 2x10- 8 653 236 rdr. 84 The inverse decay lengths for the bound wave functions were taken as a= 0. 167 F- 1 and'(= O. 36 F- 1 , and the range of nuclear interaction was chosen as a = 3. 0 F. The x2 with subscript 3H (3He) was obtained by the fit to the p - 3 H (p - 3He) angular corre la ti on at E p = 7. 8 Me V. The total X 2 turns out to be not very sensitive to R as long as it was taken to be equal or greater than 2. 0 F. For c these values of Re' the contribution J , ~from V is very much 4 14 smaller than that of J p_ from v , and as a result the angular 13 3 ' correlation becomes insensitive to the type of nucleon-nucleon potential used [cf., Section D, Part IV]. 85 REFERENCES 1. Ajzenberg-Selove, F. and Lauritsen , T., Nucl. Phys._!_!_, l (1959). 2. Austern, N . , Nuclear Forces and the Few-Nucleon Problem, Volume II, Pergamon Press, London (1960), p. 549., 3. Bacher, A. D., Ph.D. Thesis, California Institute of Technology ( 1966). 4. Balashko, J. G. and Barit, I. J., Trudy of the Lebedev Institute (Moscow, USSR) 1l• 85 ( 1965), (transl. by Consultants Bureau , New York, 1966). 5. Banerj ee , M. K., Nuclear Spectroscopy, Part B, Academic Press, New York (1960), pp . 695-731. Phys.~. 6. Barker, F. C . and Treacy, P. B., Nucl. 7. Bergman, A. A. , Isakov, A. I., Popov, J . P. and Shapiro, 33 (1962). F. L . , JETP !!_, 6 (1958) . 8. Bethe, H. A . and Bacher, R. F., Rev . Mod. Phys.~. 147 (1936). 9. Bransden, B. H., Robertson, H . H. and Swan, P., Proc. Phys . Soc. (London) A69, 877 (1956). 10 . Cerny, J ., Detraz , C . and Pehl , R . H., Phys. Rev . Lett. 15, 300 (1965). 11. Crane, H. R., Delsasso , L. A., Fowler, W. A. and Lauritsen, C. C. , Phys. Rev. 48, 125 (1935). 86 12. Demirlioglu, M. and Whaling, W~, unpublished compilation of proton stopping cross sections ( 1962). 13. deShalit, A. and Walecka, J. D., Phys. Rev. 147; 763 (1966),. 14. Donovan, P. F., Rev. Mod. Phys. ll• 501 (1965). 15. Evans, R. D., Methods of Experimental Physics, Volume V, Part B, Academic Press, New York (1963), pp. 803-806. 16. Faddeev, L. D., Mathematical Aspects of the Three-Body Problem ill Quantum Scattering, Stekov Math. Inst. Leningrad Publ. LXIX (1963), (transl. by D. Davey and Co., New York, 1965). 17. Feenberg, E., Phys. Rev. 49, 328 (1936). 18. Fermi, E., Elementary Particles, Yale University Press, New Haven, Conn. (1951), pp. 58-64. 19. Feynman, R. P., Theory of Fundamental .Processes, W. A. Benjamin, Inc., New York (1962), p . 70. 20. Frank, R. M. and Gammel, J. L., Phys. Rev. 22_, 1406 (1955). 21. Gillespie, J., Final-state Interactions., Holden-Day, Inc., San Francisco, 22. Calif., (1964). Goldberger, M. L. and Watson, K. M., Collision Theory, John Wiley and Sons, Inc., New York (1964), pp. 540-553. 23. Groce, D. E. , Ph.D. Thesis, California Institute of Technology (1963). 24. Hungerford, E. D., Gross, E. E., Malanify, J. J., Pugh, H. G • . and Watson, J. W., ORNL-4217, May, 1968. 87 25. Janecke, J., Z. Phys. 183, 499 (1965). 26. Janossy, L., Theory and Practice of the Evaluation of Measurements, 27. Clarendon Press, Oxford (1965), pp. 258-259. Jarmie, N ., Silbert, M. G., Smith, D. B. and Joos, J. S. ' Phys. Rev. 130, 1987 (1963). 28. Kavanagh, R. W. and Parker , P. D., Phys. Rev. 143, 779 (1966). 29. Kramer, P . and Moshinsky, M., Phys. Lett. 23, 574 (1966). 30. Kurepin, A. B., Trudy of the Lebede.v Institute (Moscow, USSR) ll• 1 (1965), (transl. by Consultants Bureau, New York, 1966). · 31. Landau, L. I. and Lifshitz, E. M. , Quantum Mechanics, 2nd ed., Addison-Wesley Publ. Co., Reading, Mass. (1965), pp. 562-565. 32. Lane, A. M •. .and Thomas, R. G., Rev. Mod. Phys. iQ_, 257 (1958). 3.3 . Lauritsen, T. and Ajzenberg-Selove, F., Nucl. Phys. ~. 1 (1966). 34. Lefevre, H. W., Borchers, R . R. and Poppe, C. H., Phys. Rev. 128 , 1328 ( 1962}. 35. Lovelace , C . , Phys. Rev. 135, Bl225 (1964). 36. Mani, G. S., Freeman, R., Picard, F., Sadeghi, · A. and Red on, D., Nucl. Phys. 60, 588 ( 1964}. 88 37. Mathews, J. and Walker, R. L., Mathematical Methods of Physics, W. A. Benjamin, Inc., New York (1964), pp. 361-369. 38. Maxson , D. R., Phys. Rev. 128, 1321 (1962). 39. McNally, J. H., Ph.D. Thesis, California Institute of Technology (1966). 40. Messiah, A., Quantum Mechanics, Volume I, John Wiley and Sons, Inc., New York (1962), pp. 426-428. 41. Meyerhof, W. E., Rev. Mod. Phys.}]_, 512 (1965) . 42 . Meyerhof, W. E. and McElearney, J. N., Nucl. Phys. 74, 533 (1965). 43. Meyerhof, W. E. and Tombrello, T. A., Nucl. Phys . Al09, l (1968). A. B., JETP ..!_, 2 (1955). 44. Migdal, 45. Morinigo , F. B., Ph.D . Thesis, California Institute of Tech- nology (1963). 46. Nefkens, B. M. K. and Moscatti, G., Phys. Rev. 133, Bl7 (1964). 47. Parker, P . D., Donovan, P. F., Kane , J. V. and Mollenauer, J. F., Phys . Rev . Lett . .!±· 15 (1965) . 48. Phillips, G. C., Griffy, T. A. and Biedenharn, L. C., Nucl. Phys. Q, 327 (1960). 49. Poppe, C. H., Holbrow, C. H. and Borchers, R . R., Phys. Rev . 129, 733 (1963) . 89 50. Preston, M. A., Physics of the Nucleus, Addison-Wesley, Inc., Reading, Mass. (1962), p. 179. 51. Rogers, P. C. and Stokes, R. H., Phys. Lett.§_, 320 (1964). 52. Schiff, L. I., Phys. Rev. 133, B802 (1964). 53. Seagrave, J. D., Cranberg, L. and Simmons, J. E., Phys. Rev. 119, 19 81 ( 19 6 0). 54. Spicer, B. M., Phys. Lett.±:· 88 (1963). 55. Tobocman, W., Theory of Direct Nuclear Reactions, Oxford University Press, London (1961), p. 35. 56. Tombrello, T. A., Jones, C. M., Phillips, G. C. and Weil, J. L., Nucl. Phys. 12_, 541 (1962). 57. Tombrello, T. A. and Parker, P. D., Phys. Rev. 130, 1112 (1963). 58. Tombrello, T. A., Phys. Rev. 138, B40 (1965). 59. Tombrello, T. A., Phys. Rev. 143, 772 {1966). 60. Watson, K. M., Phys. 61. Werntz, C., Phys. Rev. 128, 1336 (1962). 62. Werntz, C. and Brennan, J. G., Phys. Lett.&_, 113 (1963). 63. Werntz, C., Phys. Rev. 133, Bl9 (1964). 64. Werntz, C. and Meyerhof, W. E., Nucl. Phys. Al21, 38 (1968). 65. Young, P. G. and Ohlsen, G. G. , Phys. Lett. §_, 124 ( 1964). 66. Yu, D. U. L. and Meyerhof, W. E., Nucl. 67. ZurmUhle, R. W., Nucl. Phys. 1l:_, 225 (1965). Rev.~. 1163 (1952). Phys.~. 481 (1966). 90 Table I. The magnetic spectrometer resolutions, 68, 5q, and oE/E , used in obtaining the a-particle energy spectra from the reaction 7 Li (p,a) at eL are listed in this table [cf., Section B5 , Part II]. of E x As was indicated in Section B, Part IV , a cut-off energy = 20. 30 MeV was introduced for the least-square fit, thus . only N. data points in each spectrum were included in evaluating 1 the Xz. The spectrum marked with two stars is shown in Figure 25; while those marked with one star are shown in Figure 9. See pages 31 and 43 for more details. 91 TABLE I. The a-Particle Energy Spectra Included for the LeastSquare Fit eL ymon (deg.) (oe x o<I>) oE/E (deg. ) ( %) ]\T. ·1 2.5 909 1. Ox2 1. 11 9 5 900 1. Ox2 l. 11 9 5 1392 1. Ox4 0.56 11 1o':' 1443 1. Ox4 0.56 11 10 770 1. Ox4 1. 11 9 15'!<~< 787 1. Ox4 1. 11 10 15 2832 1. Ox4 0.56 11 20 7 51 1. 0 x4 0.56 11 20'" 2227 1. Ox4 0.56 7 · 3 o':' 1518 1. Ox4 o.56 13 30 2309 1. Ox4 0.56 7 30 1563 0. 6x4 0.56 13 40"" 2591 1. Ox4 0.56 8 45 2764 1. Ox4 1. 11 7 45 2839 1. Ox4 1. 11 8 50':' 889 1. Ox4 0.56 14 60 2710 1. Ox4 0.56 8 3430 1. Ox4 1. 11 9 70'' 6468 1. Ox4 0.56 7 80':' 1859 l. Ox4 0.56 7 90 1128 1. Ox4 0.56 7 90':' 3716 1. Ox4 l. 11 7 100':' 1389 1. Ox4 0.56 6 100 2458 O. 5 x4 1. 11 10 11 o':' 3955 1. Ox4 1. 11 5 120* 3120 1. Ox4 1. 11 5 60':' ,., 92 Table II. Some of the resonance parameters (a, ER, y z, y 2 ) p n for the first excited state of 4 He given in literature were used to fit the a-particle energy spectra summarized in Table I. The nuclear phase shifts and the scattering matrix amplitudes, appearing in expression (8), Part IV, are calculated from those resonance parameters with the expressions developed in Appendix A. See pages 44 and 7 5 for the evaluation of x . z. min 93 TABLE II. Comparison of the Resonance Parameters Given in Literature y z n (MeV) x 2. l o.o 1649 Frank et al. , 1955 20.21 1. 4 0.0 1214 Werntz, 1962 3.0 20.21 4.18 1. 50 1579 Werntz, 1964 3.0 20.31 4.44 0.50 2215 Werntz, 1964 3.0 20.41 4.77 0.50 2089 Werntz, 1964 3.6 20.21 3.02 3.00 984 Werntz, 1964 3.6 20.31 3.84 3.00 691 Werntz, 1964 3.6 20.41 3 . 52 2 . 00 615 Werntz, 1964 4.2 20.21 2.09 2.09 1098 Werntz, 1964 4.2 20.31 2.40 1. 70 789 Werntz, 1964 4.2 20.41 2.60 1.57 615 Werntz, 1964 3.0 20.36 5.20 2. 10 695 Kure pin, 1965 3.0 20.36 8.40 6.70 732 Kure pin, 1965 4.0 20.31 5.30 5.30 737 Kurepin, 1965 4.0 20.31 6.60 5.40 1441 Kure pin, 1965 4.0 20.31 10.00 8.00 2151 Kure pin, 1965 3.3 20.45 3.35 1. 74 1085 Meyerhof et al., 1965 3.3 20.41 3.53 1. 62 650 Meyerhof et al., 1965 a ER (F) 3.0 (MeV) 20.59 3.0 y z p (MeV) z min Reference 94 n( 6 n) and the scatpsx. nsx. tering matrix amplitudes D sP. taken to describe the p + 3H Table III. The scattering phase shifts 6 (n + 3 He) final-state interaction in the reaction D(3He, p) are listed here . E E is the laboratory energy of the detected protons, and p is the corresponding excitation energy of the final-state inter- x acting pair with respect to the 4 He ground state. Energies are ex- pressed in Me V, and phases are given in radians. The columns (a) are values from Meyerhof and McElearney ( 1965). The region of validity of their analysis was restricted to E S 21. 3 Me V. The x corresponding D n• 6 n and 6 n for higher excitation energies sx. psx. nsx. were obtained by extrapolations. Those columns indicated by (b) are from the resonating-group calculations of Bransden et al. ( 19 56). The inelasticity was not considered in Brans den 1 s calcula- tions, so D sP. = 1 for all the energies, and the phase shifts quoted here are the averages of those derived from Serber force and symmetrical force. For further discussion see pages 59, 66 and 67. 95 TABLE III. A Summary of the Scattering Matrix Amplitudes and the Phase Shifts used for Comparison with the Experimental Data Partial Wave E ls 8.6 8.2 7. 8 7.4 7.0 6.6 3s p E x 0 D sl 0 psi nsl (a) (a) (b) (a) (b) 20.66 20.95 21.24 21. 52 21. 79 22.06 0.77 0.66 0.63 0.62 0.62 0.62 1. 75 1. 70 1. 65 1. 60 1. 58 1. 56 - 0. 1 7 -0.22 -0.26 -0.32 -0.34 -0.37 -0.30 -0.bO -0.70 -0.82 -0.90 -0.98 -0.04 -0. 12 -0.20 -0.26 -0.30 -0.33 8.6 8.2 7. 8 7.4 7.0 6.6 20.66 20.95 21.24 21. 52 21. 79 22.06 0.94 0.85 0.77 0.72 0.67 0.62 -0.44 -0.50 -0.60 -0.65 -0.75 -0.85 -0.39 -0.48 -0.56 -0.63 -0.70 -0.75 0.22 0.50 0.67 0.85 1. 10 1. 35 -0. I 0 -0.31 -0.47 -0.57 -0.65 -0.72 lp 8.6 8.2 7.8 7.4 7.0 6.6 20.66 20.95 21.24 21. 52 21. 79 22.06 0.99 0.96 0.90 0.78 0.68 o. 57 0. 16 0.34 0.42 0.50 0.63 0.76 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.01 -0.05 -0.10 -0. 15 -0.32 -0.49 -0.00 -0.01 -0.02 -0.03 -0.04 -0.05 3p 8.6 8.2 7.8 7.4 7.0 6.6 20.66 20.95 21.24 21. 52 21. 79 22.06 1. 00 1. 00 1. 00 1. 00 1. 00 1. 00 0. 18 0.36 0.40 0.53 0.68 0.83 0.07 o. 14 0.21 0.28 0.36 0.41 o. 01 0.00 0.05 o. 10 0. 1 7 0.23 0.32 0.08 o. 13 0.20 0.30 0.40 96 FIGURE 1. Energy Level Diagram The energy levels of 4 He are shown together with all of the threshold energies and the various Q-values of the reactions that can populate the excited 4 He system. The energies are given in MeV relative to the ground-state energy of 4 He. The excitation energies involved in most of the present work were limited to the range from 19. 814 MeV, the p + 3H threshold, up to 22 MeV. The 0 + state at 20. 2 Me V was seen from the a-particle energy spectra from the reaction 7 Li(p, a) as the 1S p + 3H interaction in the final state. In Section B, Part IV, the resonance parameters for this state are deduced. tem was studied by the p - Near 21. 5 -MeV, the sys- 3 H and p - 3 He angular correlation measurements of the reaction D(3He, p). Except for the 3S n + 3He phase shifts, both the s- and p-wave phase shifts, as given by Meyerhof and McElearney, were found to be appropriate in describing the final-state interactions of the reaction. The level positions, spins , parities and the isobaric spins are taken from the latest compilation of Meyerhof and Tombrello (1968). For further discussion see Table III and pages 5, 13, 14 and 62-67. 97 FIGURE 1 ...2 ~·? 28.3 - - t2~q) 27 .8 - - - - ·- 6) ._ - - - - - - - - I D - t,==: 28.296 2n+2p ._2§ .p - - - - I ._2~ ·~ __ I I L 24 .3 _ ~2~. 111 :_, L) '-24.9{6', I} 2-, I 22.375 6u+d-a 26.072 2 H+n+p 22.4 Z-,0 21.4 o-,o 20.2 o•,o 23.847 2 H+d 19.814 3H+p 18.354 3Het-d-p 17.347 7Li+p-a 17.590 3 H+d-n 14.321 3He+t-d 12.860 3He+-\ie-2p 11.332 3H+t-2n 20578 3He+n Particle Spectra from Spectrometer at 120° 0 The protons and a-particles, with energies - 1 Me V, From the width of the proton peak, the over all electronic noise was found The energy window of the spectrometer was set at ~E page 8 . be due to the scattered particles from the wall of the spectrometer. For more details see to be 120 ke V (FWHM), thus the continuously distributed pulses below 600 ke V are believed to = E/90 - 12 ke V. kinds of particles from each other. were stopped in the counter after passing through a 5000-A nickel foil used to separate the two 200-mm 2 140..;.µ surface-barrier counter. were detected at 120 " at the focus of a 61-cm magnetic spectrometer in conjunction with a The particles produced from bombardment of a 7 Li target with 9 . 1-MeV proton beam FIGURE 2. 00 '° :J I 11 I JI -i o[ 10 30 u L...['-LI 40 JI II ~ CHANNEL t--UMBER 20 I QI 0 lri n 9530MHZ f " 201 40 r- 104 MeV E0 • lrW~~J\ 50 I I j l I I 0 QI 20 40 60r 0 20 L., 20 ~ 60 4 s 6or z r- Cf) ' - N c5- (.) MeV f = IQ065MHZ E, 1.16 10 I 1u 20 ui 30 hon rI n f 40 I = 8564MHZ CHANNEL NUMBER J ~ Ea= 0.84 Mev 1..n r L f "9.060MHZ Ect=0.9 4 MeV 50 I I I so~~--.-~~--.--~~-.-~---.~~-. 40·- 60 so~~-~~~~~--.-~~-.-~---. N M !;d c:: 0 H t-rj '° '° 100 FIGURE 3. Particle Spectra at 60° and 90° The particles emitted during bombardment of a 7 Li target on gold foil by 9.1-MeV protons were detected in a 300-µ surfacebarrier counter in the target chamber. Beam defining slits in front of the target chamber were 1. 53 mm in both vertical and horizontal directions. The angular apertures of the counter were O. 9° and 11. 7° along the 9- and qi-direction. From the spectra at 60° and 90° and s orne other angles, the actual thickness of the surface-barrier at 85-volts bias was found to be 500µ, and various groups of particles were identified. The group (A) represents the elastically scattered protons from the backing or the edge of the counter collimator and stopped in the sensitive layer of the counter. (B) represents those protons that are not stopped. is an unidentified peak. The group (C) For more details see page 10. 101 FIGURE 3 PROTON ENERGY., 9.10 MeV 15000 0.0 o.• e ~ .P~· ......J 8 = 90° - ... t .e 0 ~ 10000 t 0.- a ... ...J ~~:: ~N CID - --'° ,.·-. 5000 <:£ o.O. ~ o :Jo ...J tO 0.- t Q: ,.._ -i ~ -~ ~ t .. ..."" 0 ~ ...J "'i -b 0 0 50 100 150 0 200 250 ::l.. 0 ~ er l.O 'I- Ill. :::> t ...:i (/) z 15000 0 ~ 8 :a60° Q 10000 o.- ~ ,._:J 5000 ..... ....J!"2 ,._ 0 0 50 100 CHANNEL 150 NUMBER 200 250 102 FIGURE 4. The Target-Thickness Measurements The number of 7Li atoms per unit area of the target was measured by finding the energy loss to an a-particle group of well defined energy passing through it. In the geometry indicated, in the upper graph, the 3. 85 MeV a-particles from the reaction = 5. 00 MeV were let through the 7 Li evaporated p in the target chamber . The shift in the position of the energy cen19 F(p, n') 16 o>:' at E troid, defined by ".E.N. / ".N., was found to be 16 ± 4 keV. ,L,l 1 1 ,L,l 1 The differential cross section of the reaction 7 Li(p, a 0 ) at 30° was then found to be 1. 39 ± 0. 35mb/sr. The second method, shown in the lower graph, was carried out with a 9. 1-Me V a-particle beam from the tandem and a LiF target. The energy loss and the cor- responding 7 Li(p, a 0 ) differential cross section at 30° were respectively found to be 136 ± 6 keV and 1.88 ± 0.08mb/sr. The 7 Li target prepared in the first method may have suffered from oxygen or carbon contamination, which would lead to an underestimate of the differential cross section. details see page 11. For more 103 FIGURE 4 100 I ,f I / " .t 1 011--~~~~~~...J....~~-~-~~~---'-~~~~~~---L-~~~~~---' 3.75 3.80 3. 85 Ea 3. 90 3.95 (MeV) a --}'LiF • a-}Au o 111 a ....:: ...... (/) t- z :::> 0 0 10000 I • I 1• I ~ as 8.6 8.5 E 11 ( MeV) The Angular Distributions of 7 Li(p, ao) and 7 Li(p, a1) For the reaction additional details. See pages 12 and 13 for Li(p, a 1 ), the differential cross sections were evaluated at the a-particle energies corres- ponding to an excitation energy of 20. 06 MeV in the 4He system. 7 The The errors were estimated by "insuffi- cient11 statistics [Evans, 1963] over several repeated measurements. smooth curves serve only to connect the data points. first excited state of '1:e from. the reaction 7 Li(p, a 0 ) are shown in the laboratory system. The angular distributions of the a-particle groups leading to the ground state and the FIGURE 5. .... *" 0 FIGURE 5 105 (!) .... 0 •c ot 3PUP '°;P 0 ~ (!) .... 0 ( U/qlll) UP :op 106 FIGURE 6. The 7 Li(p, u 0 ) Angular Distribution in the C . M. System The 7Li(p, ao) angular distribution at 9. 1 MeV, along with some higher energy data due to Maxson (1962), is plotted out in the center of mass system. Since the two final-state particles are identical bosons, there is a 90° symmetry in the angular distribution. The dark circles are those data points taken at the angles corresponding to ( 1T-e CM) . The curves are the model calculations based on a PWBA and a zero-range p + 3 H interaction in the a.particle. The parameter indicated for each curve is the cut-off radius which appeared in the Butler Theory of Stripping. discussion see pages 12 and 49. For FIGURE 6 J 07 ANGULAR DISTRIBUTIONS FOR 7 . ,. LI ( p,a. He z.o Ep • 18.6MeV ~ .....• t .a t e ft .,,111~CI,, i i f9 1.0 ~ MAX&Olf(l962) 0 2.0 Ep 1115.0 Mel/ t f t .... • ..... i if I ! r 2 it 1.0 .a e bla ~ 'O .,, MAXSON (1962) 0 2.0 Ep • 9.lhhV .. .... • ..... t t I i .Q 1.0 E § .,,bl ,,ca !S.2F CENTER OF MASS ANGLE ( OE6) The a-Particle Energy Spectra at 10° and 30° Figure 8. ported by Cerny ( 196 5) , is not obvious. Except See also The third broad state at 22. 5 MeV , re- The first two peaks are For further details see pages 3 and 12. due to the first and second e x cited states of ~e. for the contamination due to 160(p, a 0 ), the spectra are quite clean . The two spectra shown here cover a wider range of the a-particle energies . FIGURE 7. ....... co 0 I l&J '"O oi; ~ b .!.. ~ I 2: ..• ~ !.o ot !5 10 1!5 2 3.!5 I C!I- PARTICLE 4.o 16ocP,a.l t ~o ENERGY (t.1eV) 4.!5 ~Hf Ba= 10° H e.s I 1 II ff jt fIf f II I I l! I /! ! ! I! IllJIJ!!ll1l I 28, 30 I '11'~ 16ocP,a.> t n Iif11" i f I I ~ f I Jrf u»ih ¥/ Ba= 30° e- PARTICLE ENERGY ( MeV) O.__~~~-'--~~~-'--~~~-'--~~~-'--~~~~~~ 3.0 3.!5 4.0 4.8 !5.0 ~!5 !5 10 '""- I~ 1! I !1!111! I 20 2!5 30 ...... -.J M ~ c: "1 Cl H '° 0 110 FIGURE 8 . Transition Probabilities The a-particle energy spectra shown in Figure 7 are plotted here as functions of the excitation energies of the recoil 'He system. The phase-space factor, which is proportional to 1 - 19. 814)] 2 , was taken out. The enhancement in the [E {E a x transition probability near the low-excitation end of the spectra was interpreted as due to the strong p + 3H final state interaction through the first excited state of the "He system. additional details see pages 13 and 29 . For FIGURE 8 111 -."'. > 1.0 CIJ :I: N' .s:J E . a:: 0 .... 0.!5 0 c:t LL w 0 t c:t a.. 16 0 ( p, a.) en ~ c:t 0L-L----=~--'-~~~_._---:L~~~--='=---J 20 :c 21 22 23 a.. ~~01~~-.----,-~-.--~~-----,.~-.-~-,----, 0 ~ I u Li.I C/) Cl) Cl) 0 a:: 0 l.!5 ...J c:t .... z Li.I a:: w LL LL Q 1.0 OJ5 20 21 22 EXCITATION ENERGY (Me V) 23 The a-Particle Energy Spectra from 10° to 120° 7 Li(p , a), are shown from 10° to 120° in steps of 10°. The arrows, corresponding The details of the fitting procedure are discussed in Section B of Part IV, and Figures 23, 24 and 25. are included in the least-square fit. to an excitation energy of 20. 30 MeV, indicate that all of the data points from those energies up of the reaction The higher-energy part of the a-particle energy spectra and the fitted spectral shapes FIGURE 9. ...... ...... N FIGURE 9 113 •o •o ..,. >Cl> •> IO ::::!: " O'i Q. II ~ a. Q) •o :::!: ;n I.LI ~><!> a:: w z w •o "" IOW rti.....1 (.) . 0 ~ . i'.= a:: ~ o' rt! " 0 CD . 0 C7I . ·g Ill N .... ........ .... ::::: --. ................. ......... - 2 0 ~ 114 FIGURE 10. Block Diagram of Slow Coincidence System The commercial instruments used, for the coincidence measurements from the reaction 7 Li(p, a.), are listed as follows: TENNELEC Model lOOA Low-Noise Preamplifier. ORTEC 410 Multimode Amplifier. Delay-line mode is used for the pulse shaping. ORT EC 420 Timing Single-Channel Analyzer (SCA). ORT EC 409 Linear Gate and Slow Coincidence. RIDL 400-Channel Analyzer. For more details, see page 15, Figures 11 and 13. FIGURE 10 115 PREAMPL PREAMPL AMPL AMPL SCA SCALER SCALER SCA 1oon LINEAR SLOW GATE COINC. r-------... DELAY GATE i-----ttNVERTa--------4 MULTICHANNEL ANALYZER SLOW COINCIDENCE SYSTEM MULTICHANNEL ANALYZER 116 FIGURE 11. The Coincidence Spectra from 7 Li(p, a) at E a = 5. 0 MeV The protons and tritons from the reaction we re detected in coincidence with the 5. 0-Me V a-particles. The a-particles were detected at 30° in the magnetic spectrome t er whos e resolutions were set a t oE/ E = 1. 11 o/o, 68 = 1 ° and oit> = 4° . The angular apertures of the triton or prot on counter in the target chamber were 3. 8° and 15. 5° respectively along the polar and azimuthal directions. The slow-coincidence resolving time was 1 µs. The dotted lines are t he kinematically predicted loci [cf., Section A, Part IIIl where the protons and tritons are expected to contribute pulses. For additional details see pages 15 and 16. FIGURE 11 117 1Li+ p~a+p+3H Ep• 9.1 MeV Ba· 30° Ea= 5.0 MeV 2 0.5 1.0 1.5 2.0 ENERGY ( MeV) 10 .5 0 124 0 120° 10 0 \ \ 116° I I I/I lo- 0 z :::> 0 112° 108° 10 0 UJ 0 z 104° UJ 0 0 z 0 0 100° 10 0 10 Jt( 96° 88~ CHANNEL NUMBER 3.0 The o. - Hand a - 3 H Angular Correlation from Li(p, a) at E 7 a = 5. 0 MeV and 8 n = 30° Because of the large 3 4 He in its center-of-mass system is assumed . For additional discussion see page 16. The curve is the geometric efficiency predicted , if an isotropic decay of the excited the uncertainty in assigning the randoms. The error bars included the statistical error and H coincidence counts was plotted here against the angular posi- tions of the counter in the target chamber. The sum of the a - H and a - kinematic (oE/88).6.8 spreading, no attempt was made to separate the protons from tritons . and subtracted by comparing with the singles [cf., Section A6, Part II] . For each of the coincidence spectra shown in Figure 11, the randoms were estimated FIGURE 12. 00 ...... ...... FIGURE 12 119 0 "' 0 "1 ( !) 0 0 w 0 m < ...J Cb 0 0 :I: ... > ii) 0. Q) cs ~ + 1' 0. + l--0--i > Q) :E 0 '°0 O'i IC) ii Q. ii 11 UJ ts Cb :J w """ a 0 en I'() 0 I{') S310U.~'1d-D 0001 /SlNOOO 30N3010NIOO 0 The Coincidence Spectra from 7 Li(p, a) at E a = 4. 1 Me V, e a = 30° The small peaks appearing from channel 110 They are randoms. The dotted lines were· drawn from the three-body kinematics. C target backing. page s 16 and 1 7 . 12 For more details see to 140 follow the two- body kinematics of the elastic scattering of the incoming protons on the The slow-coincidence resolving time was 1 µs. were, in this case, set at 7.6° and 15.5° respectively along the polar and azimuthal directions. The angular apertures of the counter in the target chamber The a-particles were detected in the magnetic spectrometer with oE/E = 1. 11 %, 08 = 1 ° and o<I> = 4°. 4.1-MeV a-particles. The protons, tritons and 3He 1 s from the reaction were detected in coincidence with the FIGURE 13. N 0 ._. 10 0 0 QI o~ ~ 2+ 0 10 20~ 0 ~ z 0 w 0 0 z :::> 20t ~ 10 :f 0 201 10 cl llllU~lc:!I 30 c:! I ...__,__, Cl c:!IC:::ld --~~:.--n nn '' / ' / .... I ' ll I I CHANNEL NUMBER 60 c:! I I I I I I 90 ~ UiL-. c I I I I I IUI I ~ ' p ' n, \ 3H n ' \ 3t-ie \ ' \ I ' \ o~~ 120 D c!:I 86° 98° 110° 122° 134' 146° 3.0 8a .. 30° d D [ Ln.......Il D 2.5 Ea•4.I MeV I C!= ENERGY (MeV) ' n 2.0 Ep=9. I MeV l.r;;,..__, I ~I ' ..... l'.5 .... c.... ~ ..... ...... rn a Li+p-a+n+3He ca c::::!l ,..r'l..ri - 0:5 7u?p-a+p+3H D c=:i I c fU::Jc::!~ 3.5 150 I I J v.i M ...... ~ c:::: () H 1-xj ...... N ...... 122 FIGURE 14. The a - H, a - 3H and a - from 7 Li(p , a} at E a 3 He Angular Correlations = 4. 1 MeV and e a = 30° After subtracting the randoms, the coincidence spectra were reduced to the angular correlation functions. Since there are more energies available in the decay of the •He system, the kinematic lines, as it was seen in Figure 13, are farther apart. It is possi- ble to separate the different groups of particles from one another. For some angles, however, the error includes the ambiguity in this separation, in addition to those from the statistics and the randoms, [cf. , the caption of Figure 12]. The positions indicated by the arrows are the angles where the final-state interactions of a + 3 H, a +Hand a + N through the relevant compound states are expected to be important. explanations see Figure 22 and pages 17 and 30. For further FIGURE 14 123 I 2C - ea = 30° E0 = 4.1 MeV Ep= 9. 1 MeV I I I 11 LIQ.1. auo.1. .j, .j, a-3H f ! 10 - t f en r t w d ,, I j::: 80 ~ f - 1 t ~ ~ ~ I 100 I 120 I 140 160 I 0 0 0 0 .j,. ! ....... a- H en 1u • ., 7L· ~4.63 2 f f- z :::> f I 01 (.) w f (.) z f ~ w 0 u z § 0 0 ~ ~ ~ 80 100 120 ilH •g I. f f ~ 80 ~ f 100 160 140 160 + a-3 He 10 140 llH•o.a. +. 0 ~ I t f 120 8 LAB ( DE.G) f 124 FIGURE 15. Stability of the Deuterated Dotriaiontane Target In the upper graph (a), the number of the 7. 8-MeV protons counted by the magnetic spectrometer, divided by the deuteron counts from D( 3He, d) in the counter set at 60° to the beam in the target chamber, is plotted versus the total integrated charge on target. The eras ses, opened circles and the dark circles are used to dis tinguish the points taken with different targets. This shows that the deuterated dotriaiontane targets are stable against the 16. 5Me V 3He beam within 10%. In the lower graph (b), the same ratio was plotted out as a function of the detected proton energies. These values were used for normalizing from one coincidence spectrum to another. For additional discussion see pages 19, 20 and 32. FIGURE 15 125 BL= 30° - Ep= 7.8 M eV 0 ,. - I.Qt- - - 8 - - 0 - - If' - - • 0 - - - - - ~- 8 - - - - - - - 8 - - - - - - - - - - - - - - II • - (/) z 0 0:: - w .,_ - :::::> w 0 I 0 0 20 I I I 40 60 80 INTEGRATED CHARGE ( µ. C) 0 0 ' en .,_ z :::::> ~o.--~~~~~...-~~~~~--~~~~~-r-~~~~~-.-~---. 0 0 BL = 30° 1.0 7.0 7.5 PROTON 8.0 ENERGY (MeV) 126 Particle Spectra at 26 °, 36 ° and 46 ° FIGURE 16. The particles emitted during bombardment of a deuterated dotriaiontane target on copper foil by 16. 5-Me V 3He 's were detected in the 300-µ surface-barrier counter (see the caption for Figure 3) in the target chamber. Beam defining slits in front of the target chamber were 1. 53 mm in both vertical and horizontal directions. The angular apertures were 2° and 8° along the 8- and ([?-direction. The spectra were complicated by the competing reactions from lZ C, such as izc( 3He, see page 21. 3 He 1 ), izc(3He, p 1 ) and 1ZC(3He, d). For more details FIGURE 16 127 1 E~ = 16.5 MeV PARTICLE SPECTRA ...,_ ~ 0 8 •46 6 ~ 5000 .r:."' -; ..- :I: .,i'="' ... ,;r:. ~ " 0 ~ .P 0 ~ -.;;- ,l=_ :r "' ~ 0 0 ~ ::i.. 75 w 6 ....... en .. - 8 :.36° "'..- ..::c 5000 .. ..;i: ~ 0 I- z :::> ~ 0 0 ,;;r. ; ,,;x ~ " 0 ,;r:.• t .,o { 0 Q) ,;r:._ "' u" :r. -!-"'~ 8 =26° ., ~ "' .:r:-_ ..,:i: :i: :c ., "' 2500 25 50 75 CHANNEL .; "' 0 .P -!- i 125 NUMBER 175 200 128 FIGURE 1 7. Angle Calibrations The loci of the angular positions of the two counters in the coincidence rneasurement of the elastic scattering , such as D + 3He-+ 3He + d, are independent of the energies involved. The lower graph shows the difference in angles of the counter in the target chamber in order to detect a deuteron or 3He in coincidence with a 3He or deuteron from the magnetic spectromete~ set at some angles along the abscissa. In the upper graph , the magnetic spec- trometer with 68 = 0. 2° , 6<I> = 4° and 6E/E = 1. 11 % was set at 37. 75° according to the reading. The dark (open} circles are the coincidences of the deuterons (3He's) from the counter in the target chamber, when at the same time the magnetic spectrometer was used to detect the 3He 1 s ( deute rons). The shift in centroids of the two groups was found to be 0. 24 ± 0.05°. As was indicated by the arrow, the actual position of the magnetic spectrometer was therefore 37. 62 ± 0. 04°. With this the absolute angle of the counter in the target chamber was known to ± 0. 1 °. page 23. For more details see FIGURE 17 129 I If Cf) I- z :::::> 0 0 0 t 0 0 f IJ.I z5 IJ.I z 0 ! f f 0 oL......L______...._______,_______--'-______.._______.________ 280 300 320 DIAL ~ READING (0.1 DEG) 2 ( !) IJ.I - Q G> ::c I') Q) 0 ----------------- ------------- ~I -2"""-=------'---------=~----......._------="-=------'-------.,..~...... 36 37 38 SPECTROMETER ANGLE (DEG) 39 130 FIGURE 18. A Block Diagram of Fast Coincidence System In addition to the electronic circuitries used for the coincidence measurements of 7 Li(p, a), an ORTEC Model 414 Fast Coincidence was inserted to the block diagram shown in Figure 10 for the D( 3He, p) coincidence measurements. solving time was 110 ns. 24 for additional details. The fast coincidence re- See the caption for Figure 10 and page FIGURE 18 13] - 0- MAGNETIC SPECTRO METER - PREAMPL PREAMPL AMPL AMPL , ... - - I SCA , SCALER SCALER SCA )C> g 1.5µ.S g DELAY · )C> ' FAST COINC. SCALER LINEAR : GATE ! SLOW COINC. MULTICHANNEL ANALYZER DELAY .-------_......, -STRETCHER,_rf..-IN-VE-~-·rc-T .... GATE 1 PROMPT GATE 1 .L_ ---,INVERT r-- FAST COINCIDENCE SYSTEM MULTICHANNEL ANALYZER The Coincidence Spectra from D( 3He,p) ate p = 30° see page 24. was 110 ns. o~ = 4° and The protons The dotted lines are drawn from the three-body kinematics. For more details The fast-coincidence resolving time The angular apertures of the counter in the target chamber were set at 2° and 8° respectively along the polar and azimuthal directions. oE/E = I. 11 %. of indicated energies were detected in the magnetic spectrometer, with 08 = 1°, coincidence spectra obtained for the energy-correlation are shown in this figure. When the counter in the target chamber was fixed at -20° with respect to the beam, the FIGURE 19. v.> ...... N 134 FIGURE 20. The Coincidence Spectra from D(3He, p) at E p = 7. 8 MeV Instead of fixing the position of the counter in the target chamber [cf., Figure 19], it was varied from 16° to 40° for these spectra at fixed proton momentum. See the caption for Figure 19 and page 24 for the resolutions of the detector, the kinematics and other de- tails. 135 FIGURE 20 136 FIGURE 21. The Energy and Angle Correlation After subtraction of the randoms and conversion to the center-of-mass of the recoil 4 He system [cf., Section A, Part III], coincidence of spectra shown in Figures 19 and 20 are shown here as an energy and angle correlation . The curves are the results of the modified Born approximation calculation. The bound-state wave functions for the deuteron, triton and 3He were taken from Yu and Meyerhof 1 s work. interaction. Rosenfeld' s force was used for the nucleon-nucleon Because there is an axis of symmetry along the recoil 4He direction, the dominant reaction mechanism was assumed to be that the incoming 3He strips its deuteron to the target deuteron and forms a final-state interacting pair of either p + 3H or n + 3 He. Following Yu and Meyerhof, a cut-off radius of 5 F was introduced in the calculation. This gives approximately the same amplitudes of the final-state interaction in p-wave ands-wave states . fitting , In the Meyerhof and McElearney' s p + 3B phase shifts and Bransden 1 s n + 3He phase shifts (cf., Table III] were used to describe the respective final-state interactions. Figure 27 and pages 25, 60 and 66. For further discussions see FIGURE 21 137 0 en 0 _J w >= 0 w ~ ..... <t _J w a:: 0. 005 8 .6 0 -16() -80 0 00 160 a 8RCM (DEG) -160 -80 '.i! 8 .6 0 0 0 0 80 ~ 160 0 138 FIGURE 22. The Velocity-Vector Diagram As it was discussed in Section A, Part III, the relative energy of the particle pair (3 + 5) or (4 + 5), from a three-body reaction l + 2 __.. 3 + 4 + 5, is identical on a cone determined by q3 and q35 or q. is the direction of the relative motion of particle i and the rest of the system, while the unit vector q.. is the • lJ The unit vector l direction of the relative motion of particles i and j. the reaction 7 Li + p ~ a + p + 3H was considered. As an example, The final state interaction of a-particle and triton through the 4. 63 MeV level in the 7 Li nucleus [cf., Figure 13] will appear at the angles where the 2 energy of the relative motion of a-particle and triton, iM 35 V 35 , is 2. 16 MeV. For more details see page 30. 139 VELOCITY VECTORS DIAGRAM 7 Li + p~a + p+ 3H Ep =9.1 MeV Ea= 4.1 M eV Ba =30 0 ~ ~ \445 .... FIGURE 22 The Chi-Squares [cf., Appendix B]. For further discussion see page 44. system is plotted around the two sets of solutions (a, ER, y 2 , y 2 ). The dotted lines inp n dicated the levels of the x 2 at which the error for each of the parameters was assigned, fit in the searching for the best resonance parameters of the first excited state of the '4He With the number of the degrees of freedom equal to 203 , the x2 of the least-square FIGURE 23. ...... 0 ~ x2 l 6051 61 x2 ' 20.32 I 5.5 2.9 3.0 20.36 / Yn2=3.03 I 5.6 ri=3.o3 ________ 20.34 , rp2 =5.3 5 I 5.4 I 5.3 ER"' 20.35 2.8 I 2.7 5.2 2.6 ER= 20.35 Yp2 =5.35 a .. 3.0 F I / 5.7 3.1 ER r.2p Yn2 x2 5 I I 600 61 x2 x2 20.40 3.1 2.0 3.2 2.1 20.44 \ 3.3 2.2 3.4 2.3 3.5 2.4 20.48 3.41 3.6 2.5 r!= 3.41 p ER"' 20.44 I ~= 0 = 4.0 F ER r,2p r; f-Ij v..> N M ~ C! 0 H ...... ,p. ,_. The 1S p + 3H Phase Shifts The arrow in- 64 for furth e r discussions. provides the phas e shifts up to this energy. See Appendix A, Figures 23 and 25 and pages 44 and dicates the cut - off energy of 20. 30 MeV in 4 He excitation energy, and the least-square fit actually respectivel y give the phase shifts by Werntz, Meyerhof et al. and Kurepin et al. The triangles, squares and circles The solid line and the dotted line are derived respectively from (3.0, 20.35, 5 . 53, 2.88) and (4.0, 20.45, 3.38, 2.23). with those given b y other authors. The phase shifts calculated from the two sets of resonance parameters {a, E , y z, y z) R p n obtained from the least- square fit described in Section B, Part IV are shown in this figure together FIGURE 24. N *"' ...... FIGURE 24 143 o'I a' 0 - I > Q) :::?: ~ 0 o• >(.'.) a:: UJ z w (.{) CJ) <l. ~ l.J.. 0 ~ ~ 0 a:: w I- z UJ 0 z 0 I- f-+-0---i 0 • 0 eo • ~ 0 0 "! 0 o~:--~~~~~~~o=--~~~~~~~o.L-~~~~~~----lo ~· I() 0 ( ~ 30) I() .l..:HHS .3S'VHd H~-d S1 a:: 0... The a-Particle Energy Spectra at 15° a = 15°. The solid line and dotted line, as they The arrow indicates the cut - off energy. Figures 9 and 24 , and page 44. and a = 4. 0 F. For more details see captions for were in Figure 24, a re respectively representing the calculations from the sets with a= 3. 0 F pared with the a-particle energy spectrum ate shape of the a-particle energy spectrum, here the corresponding calculated shapes are com- To see how different the two sets of resonance parameters are in predicting the spectral FIGURE 25. ..... .+:.+:- "C~ "C OJbl-c w E .&l L. Cl) 5 f . IrfiflHir,, If I . Ba= 1sci Ep= 9.1 MeV 7 Li ( p, a ) 4 He 20.5 t 20.0 4.8 ~2 a- PARTICLE M ENERGY ( MeV) ~ 5.6 O'--~~~L-~~~~_._~~~~----'-~~~~~L-~~~~_._....._._,__~ ·• ,, - :i! >G> - 15 21.0 4 He EXCITATION ENERGY ( MeV) N Ul M ~ ~ 1-1 1-xj Ul *"" ,_, The Triton Transfer Mechanism in 7 Li(p, a.) The dotted As was discussed in Section C, Part IV, this was still not sufficient To improve this, an interaction radius of 3. 0 F was used. . 7 The next attempt was to include the possibility that the incident proton The results are shown in the so lid curves. and the explanations for the parameters involved see pages 50-53. the system in its excited state. For more details may pick up a triton from the Li tar get forming an a-particle and leaving behind the re st of excited state of 4 He. to explain the behaviors of the angular distribution of the a-particle group leading to the first bution at E p = 9. 1 Me V. curves give the results. interaction in the a-particle, was not able to predict the forward peaking in the angular distri- In Figure 6, it was shown that the calculation, based on PWBA and a zero-range p + 3H FIGURE 26. ,j:>. O' FIGURE 26 147 --- - r' ... 0 ~ --- - - - - - - -.... -- - - - - -... -- - --- --, <!> 0 ~ Q) 0 <D ....... - 2 ........ ~ ......... ,__. ,. g ( JS/qw) ~p w en 0 0 148 FIGURE 27. The Fit to the Angular Correction from D(3He, p) at E p = 7. 8 MeV To compare the n + 3He phase shifts due to Meyerhof et al. and those due to Bransden et al. , the p - 3H and p - 3He angular = 7. 8 MeV [cf., Figure 21] were fitted with all p . the phase shifts given by Meyerhof et al. This is shown by the correlations at E dotted curves. The fits, especially to the shape of the p - 3 He angular correlation and to the branching ratio of the two modes of the reaction, were poor. The solid curves represent the same calcula- tion except that the 3S n + 3He phase shift was replaced by the value of Bransden et al. 67 and Table III. For further discussion see pages 59, 66 and FIGURE 27 14 9 1.rvv----~----,----r----~----.-------, 3 He + D --+ p + p E~_'·IG.5 MeV +3H ~ -He f Ep•7.8MeV o. I f Bp. 30° t i! -( /) f ·2 ::J t ..d ..... -- - - - - - - - ------ 0 -180 -120 -60 0 ... 60 120 180 Cf) 0 ...J w >w 3He + D -+ p + n+3He 2: .,_ E~ ."' 16.5 MeV <( .-He ...J w Ep • 7.8 MeV a: Bp = 30° ,, a -180 -120 -60 120 100