Investigation Of Two Nucleon Transfer Reactions In Fluorine-19 Using 45 Mev Protons

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Content 72-17,474 HUBER, Kent Alvin, 1937- INVESTIGATION OF TWO NUCLEON TRANSFER REACTIONS IN 9F USING 45 MEV PROTONS. University of Southern California, Ph.D., 1972 Physics, nuclear University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. INVESTIGATION OF TWO NUCLEON TRANSFER REACTIONS IN 19F USING 45 MEV PROTONS by Kent Alvin Huber A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Physics) January 1972 UNIVERSITY OF SOUTHERN CALIFORNIA T H E G R A D U A TE S C H O O L U N IV E R S IT Y PARK LOS A N G E LE S . C A L IF O R N IA 9 0 0 0 7 This dissertation, written by Kent A lv in Huber under the direction of hA-3.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R O F P H I L O S O P H Y Dean D ate February._.197_2 DISSERTATION C O M M IT T E E PLEASE NOTE: Some pages may have indistinct print. Filmed as received. University Microfilms, A Xerox Education Company TABLE OF CONTENTS Page LIST OF TABLES iv LIST OF ILLUSTRATIONS vi INTRODUCTION 1 Chapter I EXPERIMENTAL APPARATUS 3 Cyclotron and Beam Transport System Scattering Stand and Chamber Proton Beam Integration System Monitor Counter Targets Detector Systems Electronics Computer Interface Computer II EXPERIMENTAL PROCEDURE Preparation Data Acquisition Data Reduction III EXPERIMENTAL RESULTS Differential Cross-Sections for the ■*-^F (p, ^He) Reactions ii Chapter Differential Cross-Sections for the ^F(p,t) Reactions Discussion IV THEORETICAL CONSIDERATIONS Two Nucleon Stripping Reaction Theory Calculation of the Theoretical Cross- Sections for the -*-^F(p, He) Reactions Calculation of the Theoretical Cross- Sections for the -^F(p,t) Reactions V SUMMATIONS REFERENCES APPENDIX A LIST OF TABLES Table 19 3 1. Differential Cross-Section for the F(p, He) 170 Reaction to the ^70 Ground State in the Center-of-Mass System at 45.0 MeV 19 3 2. Differential Cross-Section for the F(p, He) 170* Reaction to the 0.871 MeV State in the Center-of-Mass System at 45.0 MeV 19 3 3. Differential Cross-Section for the F(p, He) 170 Reaction to the 3.06 MeV State in the Center-of-Mass System at 45.0 MeV 19 3 4. Differential Cross-Section for the F(p, He) 170 Reaction to the 3.85 MeV State in the Center-of-Mass System at 45.0 MeV 19 17 5. Differential Cross-Section for the F(p,t) F Reaction to the F Ground State in the Center- of-Mass System at 45.0 MeV 1 Q * 1 7 6. Differential Cross-Section for the F(p,t) F Reaction to the 0.500 MeV State in the Center- of-Mass System at 45.0 MeV 7. Differential Cross-Section for the ^F(p,t) "^F* Reaction to the 0.310 MeV State in the Center- of-Mass system at 45.0 MeV 19 17 A 8. Differential Cross-Section for the F(p,t) F Reaction to the 3.86 MeV State in the Center- of-Mass System at 45.0 MeV 9. Quantum Numbers Describing Angular Momentum Transfers to the Ground State of ^70 from Vector Coupling of Angular Momentum 10. Quantum Numbers Describing Angular Momentum Transfers to the Ground State of ^70 Permitted by the Selection Rules iv Page 51 52 53 54 60 61 62 63 70 71 Table Page 11. 12. Quantum Numbers Describing Angular Momentum Transfers to the 0.871 MeV State of ^0 Permitted by the Selection Rules. Optical Model Parameters Describing Elastic Scattering in the Entrance and Exit Channels 71 102 v LIST OF FIGURES Figure Page 1. Cyclotron and Beam Transport System 3 2. The Scattering Chamber 5 3. The Monitor Counter 8 4. Mean Energy Losses of Hydrogen and Helium Isotopes in Traversing 300ym of Silicon 11 5. Landau Distributions of Hydrogen and Helium Isotopes in Traversing 300ym of Silicon 14 6. Energy Losses of Hydrogen and Helium Isotopes in Traversing 300ym of Silicon 16 19 7. Limits of Investigation for F(p,t) and 19F(p, He) Experiments 18 8. Electronic Circuit for the 200ym - 2000ym Detector Telescope 20 9. Electronic Circuit for the 200ym - 1000pm - 2000ym Detector Telescope 24 10. Electronic Circuit for the 300ym - 3000ym Detector Telescope 27 11. Computer Access Channels 32 12. Two Dimensional Data Presentation of Helium Reaction Products from 45 MeV Protons Bombarding ^F at 20° 39 13. Typical Excitation Energy Spectrum for -^0 from F(p,%e) Reaction 41 171 F ( p , t ) Reaction 42 14. Typical Excitation Energy Spectrum for F from lS-p/'r, 19 1 15. Differential Cross-Section for the F(p, He) Reaction to the Ground State of -^0 55 19 ^ 16. Differential Cross-Section for the F(p, He) Reaction to the 0.871 MeV State of ^0 56 vi Figure Page 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. Differential Cross-Section for the ^F(p,^He) Reaction to the 3.06 MeV State of 0 57 Differential Cross-Section for the ^F(p,^He) Reaction to the 3.85 MeV State of 1^0 58 Differential Cross-Section for the ^F(p,t) Reaction to the Ground State of H f 64 19 Differential Cross-Section for the F(p,t) Reaction to the 0.500 MeV State of 1?F 65 19. Differential Cross-Section for the F(p,t) Reaction to the 3.10 MeV State of l^F ^ 69 Angular Momentum and Parity Assignments for the A = 17 Nuclei Schematic Description of a Pick-Up Reaction - j - j Energy Dependence of Real and Imaginary Optical Model Potentials 95 Ratio to Rutherford Elastic Scattering Cross- Sections for Nuclei Between A = 12 and A = 28 100 Theoretical Differential Cross-Sections for the 19F(p,3He) Reaction to the Ground State of ^0 104 Theoretical Differential Cross-Section for the 19F(p,^He) Reaction to the 0.871 MeV State of 1^0 105 Relative Differential Cross-Sections for Determina tion of Reaction Amplitudes for 19F(p,%le) Reaction to Ground State of HO 108 Relative Differential Cross-Sections for Determina tion of Reaction Amplitudes for 19F(p,3He) Reaction to 0.871 MeV State of Ho. 110 Theoretical Differential Cross-Section for the 19F(p,t) Reaction to the Ground State of F 117 Theoretical Differential Cross-Section for the ■*"^F(p,t) Reaction to the 0.500 MeV State of l^F 118 vii INTRODUCTION Multi-nucleon transfer reactions have provided a prolific source of experimental data for nuclear structure calculations and reaction mechanisms. These reactions, at sufficiently high bombarding energies, can be simply interpreted as the direct transfer of a cluster of nuc leons from the target nucleus. For single nucleon transfers, the theo retical description of these reactions generally utilizes the distorted wave Born approximation [1, 2, 3, 4] to predict angular distributions. Comparison of these predictions with experimental data yields spectro scopic information on nuclear structure which is independent of the kinematics of the reaction and which measures the probability that the nucleons comprising the target have a specific configuration. Transfer of two nucleons provides a means of studying the prop erties of unstable nuclei which are several nucleons removed from stable isotope targets. Unlike single nucleon transfer reactions in which the transferred angular momentum is carried by a single nucleon, these reactions involve the transfer of a pair of nucleons and many different configurations of these two nucleons contribute to the de scription of the transferred cluster. This coherence of the contribu ting amplitudes allows a test not only of the magnitude of the compon ents of the wave function but also of their relative phases [5, 12]. Such information cannot be obtained from single nucleon transfer reactions. IQ Q 19 The present experiment investigates the F(p, He) and F(p,t) two nucleon transfer reactions leading to the ^0 and ^ F mirror nuclei, respectively. Angular distributions have been obtained for transi tions leading to the ground, 0.871, 3.06, end 3.85 MeV states of "^0 and to the ground, 0.500, 3.10, and 3.86 MeV states of "^F for a bombarding energy of 46 MeV. The curves for these transitions all exhibit the characteristic forward peaking of direct interaction pro cesses Since these two reactions have identical initial states and the final nuclei are mirror nuclei, these data were analyzed assuming the final nucleus to consist of an inert ^0 core plus a single extra core nucleon. Theoretical calculations based on nuclear shell model wave functions for these configurations predict angular distributions which are consistent with the assumption of a direct interaction mechanism. CHAPTER I EXPERIMENTAL APPARATUS Cyclotron and Beam Transport System The 42" sector-focused, Isochronous cyclotron of the Southwest Regional Nuclear Consortium was used to perform this experiment. The external proton beam was obtained by passing the internal negative hy drogen ion beam through a 0.0005 inch Berillium foil to strip off two electrons. The resulting change in the charge of the ion, from -e to +e, reversed the curvature of the ions and caused the protons to pass through the fringing field of the cyclotron and into the first magnet of the beam transport system (see Figure 1). TTT PM I JJ-LI f m r IISM.1 U-U-L CYCLOTRON AND BEAM TRANSPORT SYSTEM FIGURE 1 3 At the entrance to the transport system the beam passed first through a combination magnet which collected the exiting beam and com pensated for the astigmatism introduced by the cyclotron fringing field. The first quadrupole magnet then focused the beam into a waist just beyond the cyclotron vault shielding wall. Here, a set of re motely adjustable copper slits served as the effective source for the energy analyzing part of the system. This system consisted of a 45° bending magnet and a second set of slits. A second quadrupole magnet doublet produced an image of the second slit at the target position at the center of the scattering stand. Scattering Stand and Chamber The scattering stand consisted of a central post four inches in diameter which was concentric with an eight foot diameter track. Two radial arms were supported by the center post at the inner end and by a wheel riding on the three inch wide track at the outer. Two speed electric motors were connected to the wheel to allow remote positioning of the arms. A horizontal track mounted on the top of each arm ac commodated a mounting platform with a vertical height adjustment. A remote read-out of the angular position was obtained using a servo output from an anti-backlash gear assembly located between the arm and the central post. A scattering chamber was mounted on the center post of the scattering stand by means of a ball bearing assembly. The basic vacu um chamber (shown in Figure 2) consisted of a 12" diameter aluminum TARGET ANGLE READ-OUT BNC FEED-THRU INTERNAL DETECTOR TABLE ANGLE READ-OUT INTERNAL DETECTOR TABLE SUPPORT SYSTEM LIQUID NITROGEN TRAP SUPPORT BAR INTERNAL DETECTOR TABLE DETECTOR CHAMBER EXTENDER PIPE THE SCATTERING CHAMBER FIGURE 2 6 cylinder with a 1" wide aperature extending 340° about the circum ference. The system was made vacuum tight by pinching a piece of 0.001 inch H-foil between an "0"-ring gasket fitted around the aperature and a brass retaining band [8]. The proton beam entered the scattering chamber through the front of the window and those protons which traversed the chamber with out nuclear encounters passed through the rear of the window and en tered a Faraday cup located immediately behind the chamber. Reaction products passed through a 1" diameter hole in the vertical support bar of the cylinder, through a collimated extender pipe, and into the de tector chamber. Thus, with the chamber evacuated, there was no ab sorbing material between the target and counter system. The entire system was evacuated past a liquid nitrogen trap through a pump-out located in the top of the chamber. Chamber pres sures were typically less than 20 microns of mercury. Rotation of the scattering chamber was effected by attaching the detector chamber mechanically to the scattering stand mounting platform and utilizing both the drive motors and angular read-out systems. Overall chamber dimensions allowed the investigation of laboratory angles between 10° and 170°. Proton Beam Integration System After traversing the scattering chamber the proton beam was collected in a Faraday cup consisting of a five foot long carbon 7 cylinder of four inch inner diameter and 0.5 inch wall thickness and a one inch thick carbon plate covering the far end. An electrostatic suppressor ring, located in front of the collector, was maintained at -1000 volts to prevent secondary electrons scattered from the 0.002 inch thick H-foil entrance window from entering the system. Errors due to residual gas ionization were minimized by housing both the suppressor ring and the beam collector in a brass vacuum chamber main tained at 5 x 10 ^ mm Hg. The charge was measured with a Brookhaven Nuclear Instruments, Model 1000, current integrator to within + 0.05%. Digitized output pulses from the integrator were used as a master control for all electronics. Monitor Counter During the experiment, a monitor counter was used to measure target deterioration and, at small angles where the Faraday cup could not be used, to measure the incident beam flux. The counter consisted of a piece of plastic scintillating material of sufficient thickness to stop 50 MeV protons. The plastic scintillator was optically mated to an eleven stage RCA 6655A photo multiplier tube by a one inch Lucite light pipe as shown in Figure 3. Good contact was assured at all joining surfaces by spring loading the assembly. The scintillator was preceded by a six inch collimating system. The counter assembly was placed in a lead shielding block and POWER INPUT- SCINTILLATOR LUCITE LIGHT PIPE TEST ■PULSE INPUT .UooClfl / / s / / / ~ r r ? i SIGNAL OUTPUT COLLIMATING SYSTEM PHOTOMULTIPLIER TUBE THE MONITOR COUNTER FIGURE 3 mounted on the second scattering stand mounting platform which was then fixed in angle with respect to the beam line. The particular angle chosen was the smallest scattering angle permitted by the inci dent proton flux. Targets The targets were made from commercially available Teflon (CF2 ) film of nominal thickness 0.0005 inch. The actual thickness of a target foil was determined experimentally by comparing the number of elastic scattering events occurring in the target material with the number occurring in an accurately weighed 0 . 0 0 3 inch CF2 standard. Previous experiments using Teflon targets [7, 10, 11] have shown that this material deteriorates as a linear function of the inci dent charge. This functional relationship was confirmed in the present experiment by using the monitor counter to measure the ratio of scat tered proton events to incident charge periodically during data ac quisition. The expression x = xD -(2.4 x 10-^)Q (1) O where x = actual target thickness in mg/cm , xQ = initial target thickness in mg/cm^, and, Q = charge incident on target in micro-Coulombs, yC, was obtained and used for calculating target thicknesses and life times. The useful life of a target was considered to correspond to a total incident charge of 2000 yC. Counting statistics for these cali brations were better than one per cent. 10 Four thin and one thick Teflon foils were positioned on a target ladder which rotated within the scattering chamber (see Figure 2). The two remaining target positions contained a phosphor screen to view the beam spot and a blank target frame to check for background counts. The angular position of the targets with respect to the scat tering angle could be read from a vernier scale to within +0.3 de grees. The vertical position could be set to an accuracy of + 0.06 inch. Detector Systems The choice of detector system was predicated upon the use of the Bethe-Bloche equation for particle identification. This equation relates the mean energy loss (4^0 of a particle traversing an absorbing uX medium to its charge, Zp , mass mp and total energy Ep by the ex pression [13, 14] g - V * V « b. 2.A - [la ^ _______ -»* + {«)] me Ep I (Z)(1-B2) where Z = atomic number of absorbing medium, O N = atoms/cm in absorbing medium, I = average ionization potential of atoms in absorbing medium, eZp = charge of incident particle, Mp = mass of incident particle, Ep = kinetic energy of incident particle, Be = velocity of incident particle, and, 5(B) = slowly varying function of velocity. To a first approximation, this equation predicts that parti cles incident with the same energy and charge will lose an amount of energy proportional to their mass, while particles of the same energy and mass will lose an amount proportional to the square of their charge. Since the energy loss for the latter case is greater than that for the former, particles of the same charge but different mass appear on a . 4 1 . versus E graph as a group of closely spaced hyperbola widely separ- dx ated from groups of other particles of the same mass but different charge. Figure 4 shows a typical set of mean energy loss hyperbolas for protons, deuterons, tritons, and Helium particles incident on a 300 micron silicon absorber. 22 § 2 0 - 18 in V>. ic He a . jo uj w 10 'He DEUTERONS TRITONS PROTONS 40 30 20 INCIDENT PROTON ENERGY (MEV) • MEAN ENERGY LOSSES OF HYDROGEN AND HELIUM ISOTOPES IN TRAVERSING 300 pm OF SILICON •FIGURE 4 12 Experimentally, both dE/dx and E can be measured using two solid state detectors mounted axially in a telescope type arrangement. The measurement of dE/dx is obtained from the first (or passing) de tector which is chosen thin enough to transmit all incident reaction products of interest. If, in addition, the total thickness of the two detectors is so chosen that the most energetic of these particles can be stopped in the second detector, a measurement of the total energy, E, can be obtained by summing the signals from both detectors. A detector array of this type was utilized in the present experiment. Detector Mounting The separate detectors of the telescope were rigidly fastened to a platform to maintain axial alignment. Radial alignment of the telescope with the scattering stand was accomplished by accurately machining a tongue and groove interface between the platform and the detector chamber. The platform and detector array were thermally insulated from the detector chamber by mounting two Peltier diodes beneath the tongue- and-groove surface. A current passing through the diodes produced a thermal gradient which was sufficient to reduce the temperature of the detector array to zero degrees Centigrade and thus significently re duce thermal noise and leakage current in the detectors. The noise level was further reduced by placing a piece of 0.00025 inch aluminized Mylar in front of the transmission detector to prevent illumination of the detector face by light entering through 13 the scattering chamber window. As a final precaution, the entire detector telescope was wrapped in aluminum foil to prevent any condensation on the cooled detectors. Detector Collimation Reaction products leaving the target passed through a collimator system which was designed to prevent all multi-scattered particles from entering the detector telescope. To minimize slit scattering the collimator which defined the detector solid angle was machined just 3 thick enough to stop all He and triton particles. A second collimator of slightly larger aperature and thick enough to stop 50 MeV protons was located immediately behind the defining collimator to prevent high energy protons from entering the detector system. The defining collimator had a diameter of 0.375 inch and was located by a mechanical stop a distance of 18.75 inches from the cen ter of the scattering chamber. The resulting solid angle of 3.2435 steradians permitted an angular resolution of +0.58 degree as well as a reasonable counting rate. Choice of Detectors Since the energy loss of charged particles in an absorbing med ium is a statistical phenomenon, particles of a given mass and charge and of a given energy do not all lose exactly the same amount of energy in traversing a given thickness of absorber but, rather, show a statis tical distribution about the mean energy loss predicted by the Bethe- Bloche equation. Figure 5 shows the statistical, or Landau, distri- HE HE PROTONS TRITONS 3.000 14.0 13.0 2.0CC 11 . 0 12. 0 2.500 15.0 1.500 1.000 E N E R G Y LOSS (MEV) .LANDAU DISTRIBUTIONS OF HYDROGEN AND HELIUM ISOTOPES IN 3 0 0 U m OF SILICON • .FIGURE-5 15 butions for 33 MeV hydrogen and helium particles incident on a 300 ym thick silicon absorber [13][15]. It can be seen that, if the width of the statistical distribu tions are on the order of the separation between the mean energy loss hyperbolas for particles of the same charge but different mass, par ticle identification becomes ambiguous. The Bethe-Bloche equation, however, shows that the separation between energy loss hyperbolas in creases as the thickness of absorber increases. Thus the thickness of the transmission detector in the counter telescope can be selected to provide an unambiguous particle identification between neighboring particle types. The present experiment used several different counter telescopes. A preliminary investigation to determine the feasibility of the experi ment utilized a 200 ym Silicon surface barrier transmission detector and a 2000 ym Silicon surface barrier stopping detector. The initial data were accumulated with a three detector counter telescope consis ting of 200 ym and 1000 ym, and 2000 ym Silicon surface barrier detec- 19 tors. The 200 ym detector was used as the passing counter for the F 3 (p, He) reaction and the 1000 ym detector as the stopping counter. For the ^F (p,t) reaction, the sum of the output signals from the 200 ym and 1000 ym detectors was used for the dE/dx pulse. The 2000 ym detec tor served as a stopping counter for this reaction. This choice of detectors combined two detector telescopes into a single axial arrange- 19 3 ment; a 200 ym - 1000 ym telescope for the F (p, He) reaction and a 19 1200 ym - 2000 ym detector telescope for the F (p,t) reaction. 16 The majority of the data for both reactions were collected using a 300 ym Silicon surface barrier detector and a 3000 ym Lithium drifted detector in a telescope arrangement. The thickness of the transmission detector for this configuration was an optimization between the thick ness required to identify deuterons and tritons and the thickness which would allow transmission of the Helium isotopes into the stopping detector. The dE/dx versus E graph for a 300 ym thick silicon absor ber is shown in Figure 6 for the hydrogen and helium isotopes. 24 22 20 „ 18 uj16 2 c / ) O 12 >-10 8 w 8 UJ 6 4 2 5 10 15 20 25 30 35 40 INCIDENT PROTON ENERGY (MEV) - ENERGY LOSSES OF HYDROGEN AND HELIUM ISOTOPES IN TRAVERSING 300 ym OF SILICON FIGURE 6 DEUTERONS TRITONS PROTONS 17 The Landau spread for each of these particles is indicated by the shaded area drawn about each mean energy loss hyperbola. It can be seen that this choice for a transmission detector allowed an unambigu ous identification of both the hydrogen and helium particles and per mitted the simultaneous acquisition of data for both reactions. Electronics The electronic circuitry required to perform these experiments had to perform three main functions: 1. determine that signal pulses from both the transmission and stopping detectors originated from the same charged particle, 2. provide sufficient resolution in the dE/dx signal to identify different types of particles, and, 3. provide adequate energy resolution in the E signal to permit the separation of the levels excited in the residual nucleus. The satisfactory performance of the first and second of these func tions is a necessary requirement of the detector telescope method of particle identification. The third function requires both a precise sum of the two detector signals and a pulse height analysis with high resolution. In the present experiment, the E and dE/dx signals for each re action were routed through analog to digital converters and stored in the on-line SDS computer. The size of the computer memory could only accommodate two arrays of 128 channels by 32 channels each. These 19 arrays were used to store the two E by dE/dx maps for the F (p,t) 19 3 and F (p, He) reactions. This limitation on the maximum size array which could be stored, combined with the energy resolution necessary to resolve the states in the residual nucleus, meant that the elec tronics had to also be capable of selecting only a small portion of the entire E by dE/dx map for detailed investigation. This is shown in Figure 7 in which the shaded areas indicate the limits of the arrays which were stored for each experiment. - 20 DEUTERONS TRITONS PROTONS 10 15 20 25 30 35 INCIDENT PROTON ENERGY (MEV) LIMITS OF INVESTIGATION FOR 19F(p,t) AND 19F(p,3He) EXPERIMENTS FIGURE 7 19 In practice, the functional requirements of the experiments were performed by two separate electronic systems; one a logic system which selected the detector signals of interest and the other a linear system which shaped the detector signals for pulse height analysis. While minor details of these functional circuits varied throughout the experiment the overall operation of each circuit remained unchanged. Logic System 19 3 To determine the feasibility of the F(p, He) experiment some preliminary data were collected using the logic circuit shown in Figure 8 which shows the logic system as solid lines and the linear system as dashed lines. In this circuit, which forms the basis for later additions and modifications to the logic circuit, the bipolar outputs of the linear amplifiers for the A detector (200 ym) and the B detector (2000 ym) were each routed to timing single channel ana lyzers (TSCA's). These units had a variable upper and lower level pulse amplitude discriminator and generated a constant amplitude out put logic pulse for each analog input signal which fell between the settings of the two discriminators. The logic output pulses from these units were then used to satisfy the functional requirements of the experiment. Since all charged particles entering the detectors created sig nal pulses, it was necessary to identify those signals which originated specifically from "^He and ^He particles. A tentative particle identi fication was accomplished by setting the upper and lower level dis criminators of the A - TSCA (200 y detector) such that an output logic Preamplifier Preamplifier Lin .Amo Amp _A_ AB .Sum AND ABC AND .< v AE Disc Disc Disc Lin. Gate Lin. Gate ELETRONIC CIRCUIT FOR THE 200ym - 2000ym DETECTOR TELESCOPE FIGURE 8 21 pulse would be generated only If the input signal corresponded to an energy loss in the 200 pm Silicon detector between 3 and 10 MeV. This energy range adequately brackets the mean energy losses predicted by the Bethe-Bloch equation for ^He and ^He particles having incident 19 1 19 4 energies in the kinematic range of the F(p, He) and F(p, He) re action products. Thus a logic signal from the A - TSCA signifies an energy loss in the transmission detector corresponding to that ex- O / pected for He and He particles. However, it is also possible that the input signals originated from other particles (e.g., hydrogen or Lithium isotopes) which were stopped in the 200 ym of Silicon. It was thus necessary to select those signals which originated from particles that completely tra versed the transmission detector and entered the stopping detector. This was determined by requiring a 110 nanosecond coincidence between the output logic pulses of the A and B TSCA's. Discriminator levels for the B - TSCA (2000 y detector) were set on integral; that is, all input signals greater than the lower level discriminator would generate a logic pulse. Since the thickness of the transmission detector was sufficient to stop Helium particles with less than 16 MeV of kinetic energy and the lower level discriminator of the B - TSCA was adjusted to generate logic pulses for any input signals greater than the in herent noise level of the detector, an output logic signal from the fast coincidence circuit assured that the signals being processed: (1) originated from a particle having a dE/dx in 200 ym of silicon corresponding to that of Helium particles and (2) had an incident kinetic energy greater than 16 MeV. It was now only necessary to 22 select signals having total energies corresponding to the reaction products of the ^F(p,^He) and ^F(p,^He) reactions. 3 Reaction kinematics predict He particles with a maximum kinetic energy of 35.8 MeV at 10°. Thus the upper limit on the E array was chosen to be 37.0 MeV. The lower energy limit was chosen to be greater 12 3 3 than the Q value for the C (p, He) reaction since He particles from this reaction could not be differentiated from those originating from 3 9 3 the F(p, He) reaction. The threshold for this reaction is -19.68 MeV 3 which yields He particles with maximum kinetic energies of 24.4 MeV. Consequently, the minimum kinetic energy stored was chosen to be 28 MeV. While these settings are correct for a 10° scattering angle, it should be realized that the kinetic energy of the Helium particles de creases monotonically as the scattering angle increases. At 110°, for example, the maximum energy of the reaction products is 26.8 MeV. Obviously it is not possible to obtain the entire angular distribution with a single calibration of the electronics. During the experiment different calibrations were required in each of the following angular regions: 10 to 40°, 40 to 60°, 60 to 80°, 80 to 100°, and 100 to 110°. The limitation imposed by the computer on the range of total energies which could be stored could be satisfied by several methods. The first would be to allow all E - signals, gated by the output of the fact coincidence circuit, to enter the computer memory. Pulse heights which were greater than the largest storage register (calibra ted to 37.0 MeV) would be stored in an overflow register while those which were less than the lowest storage register (calibrated to 28.0 MeV) would be stored in an underflow counter. An inherent consequence of this method was that dead-time was accumulated in both the analog- to-digital converter (ADC) and the computer for each overflow and underflow event. This disadvantage was overcome by requiring a slow coincidence between the output logic signal of the fast coincidence circuit and the output from a TSCA which generated logic pulses for E - signals between 28 and 37 MeV. The input signals for this TSCA were obtained by routing the bipolar signals from the A and B linear amplifiers through a bipolar sum amplifier. The output signal from the slow co incidence was then used to gate linear gate circuits for both the E and dE/dx unipolar signals. This particular circuit arrangement not only reduced the number of events which required analog-to-digital conversion but permitted the actual dead-time due to conversion and storage to be experimen tally determined by scaling both the number of gate pulses generated and the number of events stored in the computer. Additional scalers were used to monitor the rate at which charged particles were incident on each detector and the number of fast coincidence events. The majority of the data were obtained using a three detector telescope and the modified circuit shown in Figure 9. The design of MW 2 0 0 m m 1 0 0 0 M 2 0 0 0 |i Disc , A+B Use £' Disc C )lac i_i_£ (A+B)E'C AND Lin Lin Lin Lin Sate Gate AE I ----- l.f A+B+C wSll A AB 19F(p,3He) 19 F(p,t) ELECTRONIC CIRCUIT FOR THE 200ym - lOOOym - 2000ym DETECTOR TELESCOPE FIGURE 9 this system was based on the fact that particles with A = 3, and Z = 2 are less penetrating than particles of the same mass but having a Z = 1. 3 Thirty-five MeV He particles, for example, will lose all their kinetic 3 energy in 750 ym of Silicon while H particles require 2600 ym. The thicknesses of the three detectors in the telescope array were chosen such that the first two detectors served as the dE and (E-dE) detectors 19 3 for the F(p, He) reaction and also, by summing the output signals, 19 as the dE detector for the F(p,t) reaction. Signals from the third detector served both as a measurement of (E-dE) for the incident tri tons and as an anti-coincidence signal to reduce background counts in the (p, He) experiment. The description of this circuit is simplified if we first look at the electronics associated with the 2 0 0 ym and 1 0 0 0 ym detectors. The TSCA's for these detectors are calibrated in the same manner as for the previously described circuit. However, the logic pulses gener ated by these three units are now routed to a single fast-coincidence circuit rather that to two separate coincidence circuits. The output pulses from this single coincidence were then used as gating signals 19 3 19 for the F (p, He) linear gates. The logic circuit for the F(p,t) reaction is identical to that just described. In this case, however, the input signals do not originate (directly) from the linear ampli fier, but are taken from the outputs of both the unipolar and bi-polar 19 3 summing amplifiers of the F(p, He) experiment. This yields a dE signal proportional to the energy lost by an incident triton in traversing 1200 ym of Silicon. The TSCA's in this portion of the logic 26 3 circuit were calibrated for the H particle in a manner analogous to 3 that used for the He calibration. The output signals were routed to a single fast coincidence which generated the gating signals for 19 the F(p,t) linear gates. A second logic circuit which also allows simultaneous acqui sition of data for both reactions is shown in Figure 10. Here, the thickness of the transmission detector has been chosen to give a good 3 ^ dE signal for both He and JH particles. In this circuit, the bipolar outputs of each linear amplifiers were routed to two TSCA’s; one cali- o 3 brated for H reaction products and one for He particles. Gating pulses for each reaction were then generated using the same coincidence requirements as the previous circuit. While the logic circuitry for this latter system is functionally the same as that for the other sys tems, two changes in the handling of the detector signals must be noted. The first of these is the addition of attenuator boxes on both outputs of the linear amplifier for the transmission detector. These permitted the amplifier to be operated in a higher dynamic range and still maintain a linear summation in the summing amplifiers. The sec ond modification involved replacing the 1 1 0 ns fast coincidence cir cuit with a slow coincidence unit. This change was necessitated by the longer collection time of the 3000 ym Lithium drifted stopping detector. 300Q u 300 m A / Preamplifier Atten I --- AB Disc Disc Disc Disc Disc / V . A*B E And ABE And f- — — ► Atten AB Sum Lin. Sate A E i L J ,? F He) Data ’ ,9 F (P ( T) Data ELECTRONIC CIRCUIT FOR THE 300ym - 3000ym DETECTOR TELESCOPE FIGURE 10 Linear Pulse Svstem The linear pulse system was comprised of a preamplifier and linear amplifier for each detector, a summing amplifier to obtain a signal proportional to the total particle energy, and a linear gate and biased amplifier for both the dE and E pulses. This circuitry re mained unchanged throughout the experiment. The preamplifiers used were ORTEC, Model 105 A, charge sensi tive units. Output pulses were sent to Canberra, Model 1410, linear amplifiers to increase their amplitude and produce the pulse shapes required for pulse height analysis. The prompt bipolar pulses from both linear amplifiers were routed into the logic circuit for zero crossing timing analysis. The three microsecond delayed unipolar pulse from the dE-amplifier was routed simultaneously to the dE linear gate and to a summing amplifier where it was added to the delayed uni polar output of the (E-dE) amplifier. The output signal of the sum ming amplifier thus represented the total energy loss of the charged particle in the detector telescope and was routed to the E linear gate. Delay boxes were used as required to provide simultaneous dE and E signals at the input of the linear gates. Since the linear amplifiers had to process signals from parti cles originating from all nuclear reactions possible in the target material, the loss of resolution due to pulse pile-up had to be mini mized. This pile-up phenomenon is simply the overlap of pulses oc curring at high counting rates. It is governed entirely by the sta tistical probability of any two pulses overlapping at a given count 29 rate. During the experiment, "pile-up" counts were reduced by moni toring the output of the amplifier and adjusting the incident beam in tensity to a level where "pile-up" was negligible. System Calibration Both the linear and the logic systems were calibrated with an ORTEC precision pulser. The pulser used was precalibrated in terms of MeV/volt. The procedure used consisted of setting the AMPLITUDE control to 0.878 and adjusting the NORMALIZATION control until the 229 pulser reproduced the 8.78 MeV alpha peak from Th for each pre amplifier. The accuracy of this normalization was checked using the 6.05 MeV alpha peak from the same source. The modified pulser had a dynamic range of 1 0 volts which corresponded to effective energy losses from 0 to 40 MeV. The calibration procedure for the linear system consisted of setting the (E-dE) linear amplifier to an alpha resolution of 4 MeV per volt by observing a test pulse at the output of the amplifier. The dE - linear amplifier was then adjusted such that a single test pulse, feed alternately into the dE and (E-dE) preamplifiers, produced equal signal pulses at the output of the sum amplifier. Calibration of the system in this manner corrected for any alterations in pulse height occurring between the linear amplifiers and the summing circuit and thus assured a correct E signal pulse. Biased amplifiers were used after both linear gates to obtain a resolution of 60 Kev per channel in the energy spectrum and good 30 particle separation in the dE-spectrum. For the logic system, test pulses corresponding to the maximum energy losses of tritons and alpha particles in the transmission de tector were used to set the upper level discriminators of the dE single channel analyzers for each reaction. A similar procedure was used to set the windows for the (E-dE) and E single channel analyzers. Pulses 3 corresponding to the minimum energy losses of deutrons and He parti cles were used to set the lower level discriminators of these units. Computer Interface After passing through the linear gates the dE and E signals were routed through linear gates into separate Tullamore Victoreen Model S16 analog-to-digital converters (ADC's). Biased amplifiers were used to restrict the energy range of events into the ADC to cor respond to the lower and upper levels of the TSCA discriminators and to further amplify the signal pulses so that all pulses satisfying the discriminator settings were spread over 90% of the active range of the ADC's. Thus, for the E pulse, a signal corresponding to the lowest energy particle of interest in each experiment was amplified to 0.7 volt and signals corresponding to the maximum energy were set to 7.5 volts. The time, in microseconds, required by an ADC to convert one pulse is given by channel number t = io + ---------------- 8 31 Each ADC had one analog buffer store so that a second pulse would not be lost unless it arrived sooner than 10 p seconds after the first. A comparison of the number of gating signals prior to the ADC and the number of events stored in the computer indicated that there were no counting losses due to conversion dead time. Computer The converted E and signals from the ADC's entered the com puter memory by means of a 12-bit word. The number of significant figures, or bits, transferred for each of the signal pulses determined the size of the array into which the data were to be stored. For the 19 two F reactions, two 32 channel by 128 channel arrays were used for the — and E maps to obtain sufficient energy resolution. This was dx dE accomplished by transferring the — signal with 5 bits and the E-signal dx with the remaining 7-bits. Output pulses from a master scaler were also fed into the com puter via a priority interrupt. This scaler recorded the digitized output pulses from the beam current intregrator and, when switched 'OFF', also inactivated the remaining scalers and the computer inter face. This unit was used as a master control throughout the experi ment . The various inputs to the computer are shown in Figure 11. 32 19F(p,3He) I -------------1 AE E ADC 2 Faraday Cup i B.N.S. Model 1000 Current Integrator 19 F(p,t) AE -► ADC 3 - ► E -► ADC 4 - ► ADC 1 T S. D. S. 925 COMPUTER Magnetic Tape Master Scaler CRT TTY COMPUTER ACCESS CHANNELS FIGURE 11 The computer had two means of displaying the stored data. The first of these was an on-line cathode ray tube which continuously displayed the data being accumulated in the memory. The second means of reading the computer memory was with a tele-typewriter which could be used to dE print either the entire E by — array or any sub-array contained dx within it. This ability to print a limited amount of data in a reasonably short time was used at the end of each run to obtain statistical in formation for the ground and first excited states for both reactions. Not only could the statistical uncertainty for each level and the over all energy resolution be determined, but relative differential cross sections could also be calculated at each scattering angle investigated and used to monitor the experiment both by indicating lengths of runs to be expected and as a check on data reproducibility. CHAPTER II EXPERIMENTAL PROCEDURES Preparation The data accumulated for these reactions were obtained from five separate cyclotron runs. The set-up procedure for each of these runs consisted of first assembling the scattering chamber on the scattering stand at the approximate beam height. With the detector chamber dis mounted and the gate valve closed, the chamber was evacuated to a pressure of less than 30 microns to check the integrity of the H-foil window. A proton beam was then obtained and the height of the scat tering chamber adjusted until the beam passed through the center of the chamber. This adjustment was made by observing the target frame containing the fluorescent screen and reference grid with a closed- circuit television camera. The vertical adjustments were made by means of a turn-screw arrangement on the scattering stand center post upon which the scattering chamber was positioned. After the correct height had been attained, the detector cham ber was connected to the scattering chamber and evacuated. When the vacuum was less than 2 0 y, full bias voltage was applied to the de tectors and the calibration of the electronics completed. Simultan eously, the size of the proton beam at the target position was ad justed with the quadrupole magnets and slits in the beam transport 34 35 system. Again, this was accomplished by using the fluorescent screen and closed-circuit television camera. The typical beam spot for this experiment was 3/16" wide by 3/8" high. The energy of the external beam was determined from a previous beam energy calibration experiment [16], In that experiment, the beam energy, measured by determining the cross-over point of the angular distributions for CH2 and CD2 targets, was related to the bending mag net field strength as measured by a D2O Nuclear Magnetic Resonance (NMR) probe. According to this calibration, the beam energy during the present experiment was 45.0 + 0-2 MeV. The procedure followed to obtain this beam energy consisted in initially positioning the stripper foil to yield an energy within 1% of the desired value [17]. Fine tuning was accomplished by setting the field of the analyzing magnet with the NMR probe calibration chart and repositioning the stripper foil to obtain maximum beam current which, for this experiment, varied between 0.5 and 20 nanoampere depending on the scattering angle being investigated. Once the chamber was set-up and the proton beam focused and centered, horizontal and vertical Ni anti-scattering slits, located on the end of the beam pipe just before the entrance window, were ad justed to prevent any out-of-focus or off-energy beam components from striking the target. This particular adjustment was made by exposing glass slides until the desired collimation was achieved. The beam spot was then observed on the fluorescent screen to verify the final adjustment. 36 The final step in the set-up procedure was to calibrate the scattering stand arms with mechanical stops. After this initial cali bration the scattering angle could be read to a precision of 0 .0 0 0 2 ° with the remote read-out system. Data Acquisition At the beginning of each of the five separate cyclotron runs, measurements were made to determine the initial target thickness. Periodically throughout each of the runs, and, at the completion of each of the runs, additional measurements were made of the target thicknesses. These target calibrations were done in the following manner. The target normal was set parallel to the incident beam (e.g., a target angle of 0 degrees) and a measurement was made of the ratio of the monitor counts to the integrator counts to determine the ratio of elastic plus inelastic events per unit incident charge. A lower level discriminator was used such that only events with Q values down to 10 MeV were recorded. This was done for each of the thin CF2 targets and, at the same time, for the thick standard target. The ratio of the elastic and inelastic events occurring in a thin target foil to those events occurring in the accurately weighed, thick stan dard yielded the relative thickness of each thin target foil. A plot of the target thickness determined in this manner as a function of the total incident proton charge yielded the thickness of any target foil throughout the cyclotron run. An initial check with the blank target frame for background counts indicated there was a defocused component in the proton beam 37 which struck the target frame. Once the horizontal and vertical slits located at the end of the beam pipe were adjusted, however, a check with the same frame indicated that this component was effectively eliminated. To obtain the differential cross-section at a given scattering angle, the detector arm was remotely positioned to the angle to be in vestigated. The target angle was positioned such that the target nor mal was parallel to the detector system for target angles up to forty- five degrees. For detector angles greater than this, the target angle remained fixed at forty-five degrees with regard to the incident beam. Once these preliminary chores were completed, a proton beam was obtained in the chamber. The accumulating data was observed on the computer CRT display to verify correct setting of the various discrimi nators and biased amplifiers. Periodically throughout the data ac cumulation, the ratio of the monitor counts to the integrator counts was checked to monitor the quality of the beam spot. Also, the number of events taking place per second in each detector was monitored to assure that pile-up effects in the linear amplifiers would be negli gible. When sufficient data had been accumulated, the readings of the scalers and information identifying the scattering angle, target foil sample, and target angle were recorded in the experimental log book. The data contained in the computer memory were recorded on IBM compat ible magnetic tape for off-line analysis. Data for the ground and 1Q 19 ^ first excited states for both the F (p,t) and F (p, He) reactions were also typed out on the teletype in order that relative differen tial cross-sections could be calculated. The computer memory and all scalers were then zeroed and a run set up for the next scattering angle to be investigated. To check for reproducibility, data accumulated for angles at the start of each cyclotron run were repeated at the end of the run. 19 19 3 The data reported for both the F (p,t) and the F (p, He) reactions were reproducible within the experimental statistics for all scattering angles. Data Reduction Particle Identification: dE The magnetic tapes containing the — x E maps for the two re- dx actions were printed on an IBM System 360/75 computer at the Campus Computing Network at the University of California, Los Angeles. No 3 automatic data reduction was attempted at this time. Rather, the He and triton spectra were taken directly from the raw data maps. This 3 was accomplished by drawing a line between the He and alpha particles (or between the deuterons and tritons) to separate the different parti- cle types. A linear plot was then made of the He or triton spectrum. The experimental separation between particle types achieved with the detector telescope method is shown in Figure 12A. This fig ure shows a computer plot of total energy (vertical) versus energy loss 3 19 3 (horizontal) for He and alpha particles from the F (p, He) and 19 F (p,a) reactions. The discriminators were set to allow reaction 0.00 V EV 0.871 M EV- u • < > * « n r* » n 3.06 MEV 3.8: M EV • e T ' I — i _ ' * u f TWO DIMENSIONAL DATA PRESENTATION OF HELIUM REACTION PRODUCTS FROM 45 MEV PROTONS BOMBARDING 19F AT 20° FIGURE 12 40 products originating from this latter reaction to reach the ADC's and consequently to be stored, to insure that complete particle sepa ration was being achieved. The data shown was collected with the 2 0 0 ym - 2 0 0 0 ym detector telescope at a laboratory scattering angle of 20°. Figure 12B shows the same computer plot with binary, topo graphic contours drawn to more clearly indicate the number of parti- 3 cles. The excellent energy separation of the He particles corres- 17 ponding to the ground and first three excited states of 0 is evi- o 4 dent, as is the excellent particle separation of the JHe and He particles. While separation of particle types was unambiguous for 99% of the data, the separation of counts for the ground and first excited states often required a systematic procedure. Figures 13 and 14, 3 respectively, show typical energy plots obtained for the He and triton products at a 30° scattering angle. In order to separate spectra such as these shown, a procedure was adopted which consisted of finding a run (usually at a larger scattering angle) for which both peaks had a statistically significant number of counts. Using the data for this run a "universal" energy curve shape was determined for each of the two reaction products. The "universal" curve shape was then drawn over each of the unresolved peaks. The counts in the overlap region were divided between the two levels and an appropriated error assigned to each peak to account for the uncertainty in energy separation. New "universal" curves were determined for each cyclotron run. Number of Scattered Particles 30 20 = 33.1 com 10 0.871 0.00 3.06 3.85 Excitation Energy (MeV) TYPICAL EXCITATION ENERGY SPECTRUM FOR 170 FROM 1 9F(p,3He) REACTION FIGURE 13 Number of Scattered Particles 60 50 40 30 20 10 0.00 0.50 © com = 33.lc 3.10 Excitation Energy (MeV) TYPICAL EXCITATION ENERGY SPECTRUM FOR 17F FROM 1 9F(p,t) REACTION 3.86 FIGURE 14 4^ N> 43 Calculation of Experimental Cross-Sections Once the energy spectrum had been obtained and each state of the residual nucleus separated, the differential cross-sections for scat tering into each state was calculated using the expression dor(0 ) _ 1(0) cos 0i j A dfi I N0 t (Afi) where ID = number of incident protons, I (0) = number of particles scattered through angle 0, Afi = solid angle into which the particles are scattered, N0 = Avagadro's number, A = atomic weight of target, o t = target thickness in gm/cm , 0 = angle between the normal to the target foil and the direction of the incident proton beam. Values of the differential cross-section were calculated for all cyclotron runs for which data had been accumulated. At the same time, the relative errors associated with each measurement was also deter mined. For this calculation, contributions from uncertainties in par ticle identification, energy resolution, relative target thickness, and target angle were considered in addition to the usual statistical errors resulting from the measurement of the number of incident and scattered particles. The relative errors due to the uncertainty in the particle iden tification and energy resolution have been discussed previously. The percent error in the determination of the relative target thickness was considered to be one-third the correction as determined from Equation (1). The relative error due to the determination of the target angle, 0T, was taken as d(cos (Q))/cos (0^) or tan 0^ dOq1 for each scattering angle investigated. The total relative error was determined by taking the square root of the sum of the squares of each contributing rela tive error. These calculations were performed with a computer code specifically written for the purpose. For each cyclotron run, the initial thickness of each target foil was entered along with the values recorded for dE, E-dE, SUM, FAST coincidence, SLOW coincidence, and MONITOR scalers. Since these values depend on the target thickness and the number of incident pro tons, each scaled quantity was corrected for these variables by calculating the number of recorded events per unit target thickness and unit incident charge. The calculation of the relative cross-sections for each scat tering angle investigated were then tabulated. Here, each run was identified by a RUN number followed by data pertinent to that parti cular measurement. These data included identification of the target foil used, the actual target thickness corrected for deterioration, the laboratory scattering angle being investigated, the recorded num ber of reaction products (counts), the center of mass angle corres ponding to the laboratory scattering angle, the differential cross- section in the center of mass coordinate system, and, the relative 45 uncertainty in the center of mass value. Intermediate steps in these calculations were also printed to permit easy validation of the arithe- metic and assure correct results. A relative angular distribution was then determined by averaging the values of the differential cross-section obtained for each scat tering angle. The procedure used consisted of calculating the average of all the differential cross-section data abtained for a given scat tering angle and determining a relative error. This averaging pro cedure used the relative errors of the individual runs as weighting factors. The value of each differential cross-section was then com pared with the weighted average and those measurements which differed by greater than three times the relative error of the average differ ential cross-section for the sum of the ground plus first excited states were discarded. A new weighted average and relative error were then determined from the remaining data. This rejection procedure was considered the most relevant since the ground and first excited states were not completely resolved and the relative errors of each state separately contained an uncertainty due to the separation procedure. Rejecting data individually for each excited state would thus have depended to a large degree on the amount of uncertainty involved in the separation of the states. The method used in this experiment avoids this disadvantage and rejects data pri marily on the basis of the experimental statistics. It will be seen that the cross-sections for the second and 46 third excited states were an order of magnitude less than those for the ground and first excited states. Consequently, only those data obtained during cyclotron runs used to determine the average differen tial cross-sections for the ground and first excited states were used to determine the average values for these higher excited states. The results of this procedure for determining a relative angu lar distribution for each state excited in the residual nuclei are shown in Appendix A. These tables show the differential cross- sections and relative errors determined experimentally for each scat tering angle. The weighted average of all data at each scattering angle is shown at the end of the listing of experimental cross- sections for that angle. The third column entry for the average values, however, is three times the weighted relative error. Asterisks indicate those experimental cross-section values which differ from the weighted average by more than three times the relative error of the weighted average. For scattering angles with such values, the weighted average determined without these entries is also indicated by an asterisk. In order to obtain absolute differential cross-sections these values must be corrected for; (1) uncertainties in the measurement of the incident proton flux, (2) uncertainties in the determination of the thickness of the two mil standard target foil, (3) uncertainties in determining the solid angle subtended by the defining collimator as a result of measuring uncertainties and slit scattering, (4) counting 47 losses due to multiple Coulomb scattering in the transmission detector, 3 3 and (5) the loss of counts due to reactions between the H and He particles and the Silicon of the detector. The main contribution to indeterminacy in the integrated beam flux was due to the amount of beam which was not collected by the Fara day cup. Calculations of the multiple Coulomb scattering [18, 19, 20] of the protons in passing through the 1 mil H-foil exit window and in the air between the chamber and Faraday cup show that less than 0.5% of the incident protons were scattered into angles large enough to not be collected in the Faraday cup. To prevent secondary electron losses, the Faraday cup was fitted with an electrostatic suppressor ring. This ring also prevents electrons scattered from the entrance window from entering the col lector cup. Leakage from the Faraday cup has been measured [21] and found to be negligible. The beam current integrator used to time average the charge had an accuracy of 0.02% on all scales. Overall errors from all sources for the measurement of the incident beam current is then estimated to be less than 1%. The thickness of the 2 mil Teflon standard foil was determined from individual calculations of the weight and area. These were found to be a precision of 1% and 4.9% respectively. The total error in the cross-sections from this source is then 5%. 48 Uncertainties in the determination of the solid angle result from two effects. The first of these uncertainties is due to the errors associated with the measurement of the diameter of the defining collimator and the beam-to-collimator distance. These distances were both measured with Vernier gages to accuracies of 0.26% and 0.5%, re spectively. Additionally, uncertainties are introduced by slit edge scattering. Corrections due to this effect were calculated using the method of Burge and Smith [22] and were found to be negligible for the present experiment. Similarly, counting losses in the detector telescope due to multiple Coulomb scattering were calculated and found to be insignifi cantly small. 3 On the other hand, nuclear reactions between the incident He and triton particles and the Silicon of the detectors caused a decrease in the kinetic energy of these particles which resulted in a low energy 'tail' in each peak in the energy spectrum. The effect of this tail was to reduce the number of counts in the peaks, and to distribute those events into the energy channels below the peaks. To correct for the effects of the reaction tail, it was neces sary to add counts to each energy channel to compensate for counts lost through reactions and to subtract counts from the region below the peak in each particle distribution affected. The assumptions made for this correction were: 1. The tail was evenly distributed over all energies below the peak, 49 2. The amount of reaction tail produced by triton particles was the same as that produced by protons, and, 3 k 3. No reaction tail was produced by He and He particles. These last two assumptions are based on the data presented by Bertrand [23] on the size of the reaction tail measured for 38.6 and 60 MeV protons and for 58.8 MeV alpha particles. For the present experiment, the correction for the reaction tail amounted to 1.6% and affected only the 3.10 and 3.86 MeV states in 1?F. After correcting the data for this effect, the absolute error in the experimental differential cross-sections is 5%. I \ CHAPTER III EXPERIMENTAL RESULTS 19 3 Differential Cross-Sections for the F(p, He) Reactions 19 3 The absolute differential cross-sections obtained for F(p, He) 1 7 reaction to the ground state of x/0 are tabulated in Table 1. Table 2, Table 3, and Table 4 tabulate the absolute differential cross-sections 17 for the 0.873, 3.06, and 3.85 MeV states of 0, respectively. Figure 15 through Figure 18 show these data in graphical form. The dashed lines on these figures are included for aid in viewing only and should not be interpreted as representing theoretical pre dictions for the reaction. 50 51 TABLE 1 DIFFERENTIAL CROSS SECTION FOR THE 1 9F(p,3He) 170 REACTIONS TO THE 170 GROUND STATE IN THE CENTER-OF-MASS SYSTEM AT 45.0 MEV. ANGLE CROSS SECTION RELATIVE STANDARD DEVIATION legrees) (mmb/sr) (mmb/sr) (%) 11.08 685.05 2.03 0.29 13.84 510.58 5.96 1.16 16.60 396.53 8 . 0 0 2 . 0 1 19.36 277.13 6.79 2.45 2 2 . 1 2 190.17 2.78 1.46 24.87 163.70 4.28 2.61 27.62 133.61 4.24 3.17 30.36 126.92 3.47 2.73 33.10 130.60 2.53 1.93 35.83 117.25 8.47 7.22 38.56 97.60 0.23 0.24 41.27 69.00 2 . 2 1 3.20 43.98 51.16 2.64 5.16 46.69 36.83 6.13 16.64 49.38 31.29 0.74 2.36 52.07 30.54 3.07 10.05 54.75 29.74 0.54 1.81 57.42 27.34 2.41 8.81 60.08 29.78 0 . 6 8 2.28 65.37 27.72 1.16 4.18 70.62 19.66 0.97 4.93 75.83 12.83 0.24 1.87 80.99 9.54 0.29 3.03 8 6 . 1 1 9.56 0.37 3.87 91.19 12.63 1.41 11.16 96.21 11.62 0.23 1.97 106.11 5.49 0.56 1 0 . 2 0 52 TABLE 2 DIFFERENTIAL CROSS-SECTION FOR THE 1 9F(p,3He) 170 * REACTION TO THE 0.871 MEV STATE IN THE CENTER-OF-MASS SYSTEM AT 45.0 MEV. ANGLE CROSS SECTION RELATIVE STANDARD DEVIATION (degrees) (mmb/sr) (mmb/sr) (%) 11.08 194.00 13.42 6.9 13.84 84.63 2.15 2.5 16.60 63.87 1.95 3.1 19.36 72.34 2.06 2 . 8 2 2 . 1 2 82.61 4.31 5.2 24.87 95.24 7.17 7.5 27.62 81.71 4.24 5.2 30.36 74.25 3.78 5.1 33.10 67.65 2 . 6 6 3.9 35.83 46.09 4.63 1 0 . 0 38.56 32.97 1.33 4.0 41.27 19.03 2.42 12.7 43.98 16.64 0.95 5.7 46.69 1 2 . 0 1 0.93 7.7 49.38 12.74 0.50 3.9 52.07 17.74 2.29 12.9 54.75 18.11 0 . 6 8 3.8 57.42 17.89 1.91 10.7 60.08 17.05 0.46 2.7 65.37 10.13 0.23 2.3 70.62 4.17 0.51 1 2 . 2 75.83 2.57 0.33 1 2 . 8 80.99 3.06 0.63 2 0 . 6 8 6 . 1 1 4.38 0 . 1 2 2.7 91.19 4.33 0.09 2 . 1 96.21 3.39 0.09 2.7 106.11 0.71 0.18 25.4 53 TABLE 3 DIFFERENTIAL CROSS-SECTION FOR THE 1 9F(p,3He) 170 REACTION TO THE 3.06 MEV STATE IN THE CENTER-OF-MASS SYSTEM AT 45.0 MEV. ANGLE CROSS SECTION (degrees) (mmb/sr) 11.13 45.28 16.68 24.41 19.45 15.58 2 2 . 2 2 9.79 24.98 4.45 27.74 7.72 30.50 9.15 33.25 10.17 35.99 8.56 38.72 6.94 41.45 4.01 44.17 4.17 46.89 5.20 49.59 4.52 54.98 2.73 57.66 2.94 60.63 0.67 70.89 1.65 76.11 0.26 81.28 1 . 0 2 86.40 0.57 91.48 1.43 RELATIVE STANDARD DEVIATION (mmb/sr) (%) 9.16 2 0 . 2 2.35 9.6 1.51 9.7 2.24 22.9 0 . 6 6 14.8 1 . 1 2 14.5 1 . 1 1 1 2 . 1 1 . 0 2 1 0 . 0 1.40 16.4 1.13 16.3 0.63 15.7 0.39 9.4 0 . 6 8 13.1 0.26 5.8 0.27 9.9 0.74 25.2 0.56 83.6 0.35 2 1 . 2 0.30 15.4 0.24 23.5 0.28 49.1 0.15 10.5 54 TABLE 4 DIFFERENTIAL CROSS-SECTION FOR THE 1 9F(p,3He) 1 70* REACTION TO THE 3.85 MEV STATE IN THE CENTER-OF-MASS SYSTEM AT 45.0 MEV. ANGLE CROSS SECTION (degrees) (mmb/sr) 11.13 19.87 16.68 24.55 19.45 21.46 2 2 . 2 2 20.27 24.98 14.55 27.74 13.70 30.50 9.84 33.25 5.53 35.99 1.95 38.72 2.70 41.45 6.06 44.17 6.15 46.89 5.34 49.59 4.53 54.98 4.54 57.66 1.83 60.32 2.16 65.63 2.34 70.89 2 . 1 2 76.11 0.74 81.28 2.72 86.40 1.26 91.48 0.77 RELATIVE STANDARD DEVIATION (mmb/sr) (%) 6 . 0 2 30.3 1 . 8 8 7.6 2.82 13.1 1.61 7.9 0.92 6.3 1.30 9.5 1.37 13.9 0.70 12.7 1.53 78.5 0.52 19.3 1 . 6 6 27.4 0.38 6 . 2 0.92 17.2 0.32 7.1 0.63 13.9 0.58 31.7 0.51 23.6 1.05 44.9 0.56 26.4 0.69 93.2 0.40 14.7 0.40 31.7 0.32 41.6 d if f e r e n t ia l cross section O u b / s t e r ) 55 1000 100 40 60 80 100 1 2 0 © C M DIFFERENTIAL CROSS-SECTION FOR THE 19F(p,3He) REACTION TO THE GROUND STATE OF 170 FIGURE 15 differential cross section (JUB/STER) 56 200 10 0.3 20 40 60 80 (00 (20 ® CM 19 3 'DIFFERENTIAL CROSS-SECTION FOR THE F(p, He) REACTION TO THE 0.871 MEV STATE OF 170 FIGURE 16 DIFFERENTIAL C R O S S S E C TIO N (JUB/STER) 57 100 20 40 60 8 0 100 120 ©CM DIFFERENTIAL CROSS-SECTION FOR THE 19F(p,3He) REACTION TO THE 3.06 MEV STATE OF 17 0 FIGURE 17 DIFFERENTIAL C R O S S SECTIO N (JJB/STER) 58 100 10 20 40 60 80 100 120 ® CM DIFFERENTIAL CROSS-SECTION FOR THE 19F(o,3He) REACTION TO THE 3.85 MEV STATE OF I70 FIGURE 18 59 19 Differential Cross-Sections for the F(p,t) Reactions 19 The absolute differential cross-sections obtained for F(p,t) 17 reaction to the ground state of F are tabulated in Table 5. Table 6 , Table 7, and Table 8 tabulate the absolute differential cross-sections for the 0.500, 3.10, 3.86 MeV states of ^0, respectively. Figures 19 through 21 show these data in graphical form. Again, the dashed lines on these figures are included for aid in viewing only and should not be interpreted as representing theoretical predictions for the reaction. 60 TABLE 5 DIFFERENTIAL CROSS-SECTION FOR THE 19F (p,t)17F REACTION TO THE 17F GROUND STATE IN THE CENTER-OF-MASS SYSTEM AT 45.0 MEV. ANGLE CROSS-SECTION (degrees) (iranb/ sr) 13.90 538.47 16.67 504.56 19.44 451.63 2 2 . 2 1 361.60 24.97 262.31 27.73 205.99 33.23 181.10 38.71 159.44 41.44 139.75 44.16 115.60 46.87 96.54 49.57 70.31 52.27 68.55 54.95 60.33 60.30 43.24 65.60 41.20 70.86 37.31 76.08 31.88 81.25 20.74 86.37 14.95 91.45 13.44 96.47 11.94 106.38 4.48 RELATIVE STANDARD DEVIATION (mmb/sr) (%) 9.85 1.82 18.75 3.71 30.11 6 . 6 6 17.14 4.74 27.79 10.59 18.22 8.84 11.15 6.15 7.11 4.45 4.14 2.96 9.27 8 . 0 1 6.81 7.05 2.47 3.51 4.90 7.14 2.90 4.80 1.81 4.18 2.39 5.80 1 . 8 8 5.03 8.14 25.53 1.83 8.82 1.46 9.76 1.72 12.79 2 . 0 1 16.83 0.52 11.60 61 TABLE 6 DIFFERENTIAL CROSS-SECTION FOR THE 19F (p,t) 1 7F* REACTION TO THE 0.500 MEV STATE IN THE CENTER-OF-MASS SYSTEM AT 45.0 MEV. ANGLE CROSS-SECTION (degrees) (mmb/sr) 13.90 146.84 16.67 77.35 19.44 109.25 2 2 . 2 1 155.10 24.97 231.27 27.73 234.54 33.23 207.08 38.71 113.73 41.44 60.71 44.16 35.57 46.87 10.67 49.57 10.47 52.27 19.56 54.95 29.23 60.30 45.02 65.60 33.25 70.86 18.28 76.08 6.05 81.25 2.40 86.37 5.57 91.45 7.45 96.47 7.00 106.38 1.91 RELATIVE STANDARD DEVIATION (mmb/sr) (%) 14.36 9.8 6 . 2 0 8 . 0 3.84 3.5 10.03 6.5 16.89 7.3 7.03 3.0 15.53 7.5 6.99 6 . 1 4.61 7.6 1.73 4.9 2 . 6 6 24.9 0.71 6 . 8 2.57 13.1 0.76 2 . 6 0.47 1 . 0 1.26 3.8 1.15 6.3 0 . 1 2 2 . 0 0.59 25.6 0.41 7.4 0.43 5.8 0 . 0 2 0.3 0.33 17.3 62 TABLE 7 DIFFERENTIAL CROSS-SECTION FOR THE 19F (p,t) 1 7F* REACTION TO THE 3.10 MEV STATE IN THE CENTER-OF-MASS SYSTEM AT 45.0 MEV. ANGLE CROSS-SECTI (degrees) (mmb/sr) 16.76 39.20 19.55 39.96 22.33 11.19 27.88 23.49 38.90 18.68 41.64 10.53 44.38 8.26 49.82 3.14 60.58 3.29 65.90 5.34 71.18 4.40 76.40 2.91 81.58 0.43 86.71 0.81 RELATIVE STANDARD DEVIATION (mmb/sr) (%) 5.24 13.4 6.85 17.1 3.44 30.7 0.81 3.4 3.57 19.1 0.58 5.5 0.80 9.7 0.69 2 2 . 0 0.69 2 1 . 0 0.93 17.4 0.09 2 . 0 0.54 18.6 0.26 60.5 0 . 2 1 25.9 63 TABLE 8 DIFFERENTIAL CROSS-SECTION FOR THE 19F (p,t) 1 7F* REACTION TO THE 3.86 MEV STATE IN THE CENTER-OF-MASS SYSTEM AT A5.0 MEV. ANGLE CROSS SECTION RELATIVE STANDARD DEVIATION (degrees) (mmb/sr) (mmb/sr) (%) 16.76 43.57 7.38 16.9 19.55 28.00 5.56 19.9 27.88 2 2 . 0 1 4.18 19.0 38.90 13.70 3.72 27.2 41.64 13.42 2.45 18.3 44.38 15.42 0.54 3.5 49.82 10.36 0.91 8 . 8 71.18 6.59 1.07 16.2 76.40 5.65 0.77 13.6 81.58 5.43 0.89 16.4 86.71 2.77 0.51 18.4 DIFFERENTIAL C R O S S S E C TIO N (jJB/STER) 64 I O O O 100 20 40 G O 80 100 120 ©CM DIFFERENTIAL CROSS-SECTION FOR THE ^F(p,t) REACTION TO THE GROUND STATE OF 'f FIGURE 19 DIFFERENTIAL C R O S S S E C T IO N (JJb/STER) I O O O 100 20 40 60 80 100 120 65 © CM DIFFERENTIAL CROSS-SECTION FOR THE 19F(p,t) REACTION TO THE 0.500 MEV STATE OF 17p FIGURE 20 d iffe re n tia l cross section (pb /s te r) 66 100 * \ 20 40 60 8 0 100 120 ©CM DIFFERENTIAL CROSS-SECTION FOR THE 19F(p,t) REACTION TO THE 3.10 MEV STATE OF 17F FIGURE 21 67 Discussion IQ * 3 The F(p, He) Reaction The ^F(p,^He) ^0 and ^F(p,^He) "^0* reactions have been in vestigated at lower energies by several authors. Dickens et al [10, 36] used 30 MeV protons incident on a Teflon (CF2 ) target to study the 17 ground and first excited state of 0. While the angular distribu tions found for these transitions indicate the overall structure, the measurements did not resolve the details of the present data. The University of Minnesota Group [24] also used Teflon foils to study the ground and first three excited states at 0.871, 3.06, and 3.85 MeV for 40 MeV protons. When corrected for the expected kinematic effects, these data exhibit the same structure as presented in the previous section. All experimental angular distributions exhibit the forward peaking characteristic of nucleon transfer reactions. In the bom barding range considered, no compound nucleus contributions were anti cipated and the lack of symmetry of the data about 90° center of mass indicates that this is indeed the case. However, it must be noted that the shape of a distribution does not definitely determine the reaction mechanism. A better approach is to measure the polarization of the outgoing particle. Unfortunately, such data is unavailable for the reactions studied in the present work. The total cross sections of the ground and 0.871, MeV excited state have been determined by numerical integration of the differen tial cross-sections over angle and are 1.855 millibarn and 0.66 milli- 68 barn, respectively. For the 30 MeV data by Dickens, et al, these partial cross-sections are 1.00 millibarn and 0.48 millibarn, respec tively . Of the various possible direct reaction mechanisms, a pick-up of a deuteron cluster from the target appears the most attractive. Then the successes of single nucleon transfer analyses to predict the angular momentum transferred lead one to expect analogous predictions with the transfer of a cluster comprised of two nucleons. In the two nucleon process, however, the angular momentum transferred by the cluster is a result of the vector coupling of the angular momenta of each individual nucleon. Figure 22 shows the angular momentum and parity assignments for the ^0 nucleus. Realizing the ^ F nucleus has an assignment of l/2+, we can determine the possible angular momentum carried by the trans ferred cluster by applying the angular momentum selection rules [1,5,12]; |Ji — Jf| £ J £ |j^ + Jf| sa “ sbl £ s £ lsa + sbl where is the angular momentum of the target nucleus, Jf is the angular momentum of the residual nucleus, Sa is the spin angular momentum of the incident particle, and, Sjj is the spin angular momentum carried off by the light reaction product. 3/2 - 5/2 - 1/2 - ■4.55 3.85 • 3.06 1/2 + 5/2 + ,0.871 , 0.00 170 3/2 - 5/2 - 1/2 - -4.69 -3.86 -3.10 1/2 + 5/2 + -0.50 '0.00 17. ANGULAR MOMENTUM AND PARITY ASSIGNMENTS FOR THE A = 17 NUCLEI FIGURE 22 70 The spin of the incident proton is 1/2 and that of the %e particle is also 1/2 so the spin transfer is either 0 or 1. To determine the 19 + total angular momentum transfer we note that the spin of F is 1/2 17 . + and that of 0 is 5/2 which yield values of J = 2 or 3 for the trans ferred cluster. Using the vector coupling of angular momentum, we then find that the cluster may have the values of the quantum numbers shown in Table 9. TABLE 9 QUANTUM NUMBERS DESCRIBING ANGULAR MOMENTUM TRANSFERS TO THE GROUND STATE OF 170 FROM VECTOR COUPLING OF ANGULAR MOMENTUM J = I L + S | ________________L________S_____ J 2 = L + 0 2 0 2 2 = L + 1 1,2,3 1 2 3 = L + 0 3 0 3 3 = L + 1 2,3,4 1 3 However, since there is no parity change, those combinations containing an odd angular momentum exchange cannot occur. The possible quantum numbers for a transition to the ^O ground-state are then shown in Table 10. TABLE 10 QUANTUM NUMBERS DESCRIBING ANGULAR MOMENTUM TRANSFERS TO THE GROUND STATE OF 170 PERMITTED BY THE SECTION RULES L S J 2 0 2 2 1 2 2 1 3 4 1 3 Through similar arguments one can determine that the possible transitions to the l/2+ ^ 7 0 first excited state are described by the quantum numbers shown in Table 11. TABLE 11 QUANTUM NUMBERS DESCRIBING ANGULAR MOMENTUM TRANSFERS TO THE 0.871 MEV STATE OF 170 PERMITTED BY THE SELECTION RULES L S J 0 0 0 0 1 1 2 1 1 An inspection of the experimental angular distributions, indicates that, while the 0.871 MeV state is predominately L = 0, the ground state is most probably comprised of both L = 2 and L = 4 contributions. 72 For the 3.06 MeV and 3.85 MeV states, similar arguments lead to orbital angular momentum transfers of L = 1 and L = 1 or 3, respect ively . 19 The F(p,t) Reaction 17 5 ^ The ^F(p,t) 170 and ^F(p,t) 0 reactions have also been in vestigated at 40 [25] and 30 [26] MeV. Dittman et al studied the ground and 0.500 MeV state while Reynolds et al obtained data for these states and the 3.10 and the 3.86 MeV levels as well. Again, except for kinematic compression, these data show excellent agreement with the present data. Holmgren and Fulmer [7] obtained partial cross-sections 19 for the F(p,t) reaction at 22.8 MeV incident proton energies. This experiment was able to resolve the ground state and the 0.500, 3.10, 17 3.86, and 4.69 MeV states of F. The resolution was such that no structure was found in the angular distributions for states above 0.500 MeV. Also, the distributions are distinctly more symmetric about 90° indicating a more significant compound-nucleus, contribution. The partial differential cross-sections for the 22.8, 30.0, and 46 MeV ground state reactions are, respectively, 1.80 mb, 1.57 mb, and 2.58 mb. For the transition to the first excited state, we find 1.5 mb 1.55 mb, and 1.56 mb. For these reactions the transferred "di-neutron" cluster has S = 0 and application of the angular momentum selection rules reduce to IJi “ JfI 1 L 1 IJi + JfI 73 This leads to 'pure' angular momentum transfers of L = 2, L = 0, L = 1, and L = 3 for transfers to the ground and first three excited states. The experimental angular distributions are indeed consistent with these expected angular momentum transfers. The effect of the higher angular momentum transfer contributions 19 3 to the F(p, He) reactions can now be estimated by comparing the angu lar distributions for these reactions with the results obtained for the ^^(p,t) reactions in which only a single value of angular momen tum can contribute. For the ground state reaction, the L = 4 contribu tion to the L = 2 angular distribution causes more pronounced inflec tion points in the curve and produces an additional maximum in the region of 90° center-of-mass. The contribution of an L = 2 angular momentum transfer to the 0.871 MeV excited state, L = 0 transfer effectively shifts the angular distribution 5° to 8° toward larger angles and fills in the valleys. No definitive conclusions can be reached for the 3.06 and 3.85 MeV states of "^0. •CHAPTER IV THEORETICAL CONSIDERATIONS Present efforts in the theory of nuclear reactions are devoted to attempts at finding simple models that will predict as many experi mental observations as possible. Two of these models, the compound nucleus model and the direct interaction model, have proven particu larly successful. The first of these reaction models assumes that the reaction takes place in two distinct steps. For the first step, the target nucleus captures the incident particle to form a many body system of strongly-interacting particles called a compound nucleus. In such a system, the incident particle has a short mean free path and shares its energy with the other particles of the system so that it cannot be re-emitted until, as a result of further exchanges, sufficient energy is again concentrated on this or a similar particle. This second step, i.e., the break-up of the compound nucleus into different reaction products, is determined only by the properties of the compound nucleus and is independent of its mode of formation. The time required to obtain such a state of equilibrium can be estimated from the uncertainty principle and the experimentally determined energy level widths as; 74 75 AE At ~ h (1.0 ev) At ~ 10-^ ev-sec or -15 At = 10 seconds When the capture process is complete, the compound nucleus is in a state of complete equilibrium in which there are no preferred directions. Thus this model predicts angular distributions of emerging particles which are symmetric about 90° center-of-mass. The compound nucleus hypothesis gave an excellent account of many diverse types of nuclear reactions. However, certain features not easily described by the theory emerged as bombarding energies increased above 10 MeV. Predominent among these were the emission of excess particles of high energy in comparison with the number expected ac cording to the compound nucleus theory and a pronounced peaking of the experimental angular distributions toward small scattering angles. These differences were explained by hypothesizing a direct interaction process in which the energy and momentum of the incident particle were transferred directly to a few nucleons of the target. Such a reaction mechanism is not inconsistent with the compound nucleus hypothesis since the latter requires small mean free nucleon paths only in the compound nucleus system and such an assumption is not necessarily true for the first interaction of the incident particle with the target nucleus. Theories of direct interactions assume this first collision occurs with a nucleon near the surface of the target nucleus. If a mean free path of nuclear dimensions is assumed, the 76 struck nucleon may emerge without the formation of a compound nucleus. The time required for a direct reaction can be estimated as the time it takes the incident projectile to traverse the target. This is — 91 approximately 10 seconds, a time significantly shorter than that required for the formation of a compound nucleus. It is obvious that these two reaction models are just extreme descriptions of what actually occurs in a nuclear reaction and it would be more correct to say that the direct interaction aspects of a reaction are those which involve only a few of the many degrees of freedom of a nucleus. The minimum number of degrees of freedom neces sary to describe a reaction are those which describe the initial and final states. The direct interaction model studies these and no others. The remaining part of the wave function involves complicated excitations of many degrees of freedom and describes the compound nucleus effects. Two Nucleon Stripping Reaction Theory In this section will be presented the general derivation of the differential cross-section for two particle transfer reactions. A complete description of the Born approximation on which this deriva tion is based is beyond the scope of this work and the reader is re ferred to the excellent formulations of Glendenning [2,5,28], Satchler, [4] and Towner and Hardy [1]. Due to the nature of the experimental work, it will be convenient to discuss a pick-up reaction such as that shown schematically in Figure 23 but analogous expressions can also be derived for stripping processes. J -** + © = > ® + ( © f e ! r* k b A SCHEMATIC DESCRIPTION OF A PICK-UP REACTION FIGURE 23 For this pick-up reaction, A(a,b)B, formal scattering theory [3, 29] yields the following exact expression for the transition amplitude t = < u r K A - U a * 1 # r > (2) where is the sum of all two body interactions between each nucleon in the projectile, a, and those in the target nucleus, A = (B+l+2), and Ua^ is the optical potential describing elastic scattering in the incident channel. The difference between these two potentials is that portion of the interaction which enters into the two-nucleon transfer process. 78 r ~r i-J Since the wave function X j . of the total Hamiltonian is not known, this expression is evaluated under the Born approximation which replaces this wave function with , a solution of the Schrodinger equation for elastic scattering in the exit channel. These elastic scattering wave functions may be expanded in terms of relative and internal coordinates: $ T = K p W ) (3a) " l T dfc / \ I t") . % = ® Mt ( ° ? r‘ ^ k j a±^ <3b > where for the sake of definiteness, we have assumed a (p,t) reaction. The various constituents of these wave functions are: -Tp (2..0X) “ wave function of target nucleus ~ wave f u n c t i ° n residual nucleus, - spin wave function of incident proton, j - "internal" wave function of triton, and (X> ( n ~ i ) ~ elastic scattering wave function of light nuclei in either the proton channel, p, or the triton channel, t. Two approximations are commonly used to describe the elastic scattering wave functions; these are the plane wave and the distorted 79 wave approximations which are widely referred to in the literature as the PWBA and the DWBA methods, respectively. In the PWBA, the wave functions describing the relative motions between the light particles and the heavy nuclei are found by neglecting the interaction between the two parts. They thus satisfy a field free Schroedinger equation and are therefore plane waves; i.e., PWBA leads to the Butler formula for the angular distribution. This method has been used extensively as a means of extracting spectro scopic information from stripping reactions due primarily to the simpli fications the method introduces in the calculations. With the avail ability of digital computers, however, the more exact DWBA has become the general technique for determining this same information. This method accounts for the scattering and partial absorption of the inci dent deuteron and outgoing proton by the nucleus. These distortion effects can be simulated in the PWBA by introducing a cut-off radius which, in effect, causes absorption of the lower partial waves. How ever, no account of distortion of the higher partial waves, which are of importance for pick-up reactions, is considered in this approxi mation. (4) The determination of the differential cross-section with the The DWBA method for taking into account the scattering of the incident and outgoint particles by the nuclear field was first formu lated by Tobocman. In this case the wave functions ( j ) ^ and < | ) ^ t 80 satisfy Schroedinger equations containing the interaction of these particles with the nucleus, and they are therefore distorted from the field-free plane-wave solutions of Equation (4). The interaction potential which appears in the transition amplitude can be written more explicitly as: 3 Va f t -U,A = < V i - ) h - Z < A k - U aa) (5) J = ' where V^a is the two body potential between nucleons j and the inci dent probe, a. The sum j is over all nucleons in the residual (core) nucleus. Under the assumption that a core-independent transition will dominate over a mechanism that first excites and then de-excites to the core nucleus, we ignore the second term of the interaction poten tial. Then, the transition matrix can be written as Because of the physical picture we have of the pick-up process, we expand the wave function for the initial nuclear state on a basis exhibiting the residual nucleus and the transferred pair, a t * jlA\ where A* is the parentage factor connecting the nucleus (A + 2) to (A) and represents the degree to which the ground state of (A + 2) 81 has as its parent the state of the nucleus (A) that is formed in the interaction plus two nucleons in the state 6 = n^£^s^ , n2&2s2 » j> m . is the wave function of the transferred two particle system, and is the wave function of the nucleons comprising the core of the target nucleus. Expanding Eq. (7): Mi 1 3M jm 0,1*1 j2*t Mv JM 0,*i 01W1 f i t'll This expansion has assummed jj coupling. For subsequent convenience, however, it is desirable to separate the orbital and spin parts of the pair wave function by transforming it into LS coupling. This is accomplished by means of the following expression: (1) ^— i < a { s.crT I j i < * i, J 2. I j • * » y 4*^, ^ X ’ H 1 . r s -i |LA>* < T Z .0, OZ CT ) Av m A t I , S, | .S2 * <s1<r, s,«i'|sr><LA3£lo“'> ( 8) 82 where j = ^2j+l* etc. Substituting this transformation we have ■f3’ 1 - Z / V I ' < 3M jm jdi -O, 5 , 0 1 t- A 5£ *<ZAssr|>>> ($)<}* (a,) * f < • > > A A L s, 5t S j . o’z. j -I (9) where J.Jt L s,5, S ji ji 3 = yzL+r ]izs^ / ^ 5i m and the bracket is a 9-j symbol, Substituting y into the transition amplitude yields “ Mi T~ ^ A | J‘ i ML> < ^ ,A A l |t.AX^o'l5,rl |s2> j>jv £,\ <UA5S iv3 <-x, | s ? . X Z W + > > ^ ( r t ‘ ) r r, W ) Z ‘ C*'>1 + r (B f } > r, ( IO) 83 where we have performed the summations over J and M after first noting that the overlap states that the core nucleons retain the same configuration as in the target nucleus; i.e., there is no core excitation. Expanding the wave function of the triton in a similar manner, we find ( M-t Lt/\t Sfcflt VoysV' s'p^'sV1 |s-t£fc > s,'< s l K < s,V,'s,:<u |s'<r'> (i d With this expansion, the spin overlap in the transition amplitude is < < = ? * 2 <LfcAt St 0^ | JtM t > SC(W ) I (01 C ) N Mfc f fl rP Ur/VSv ' r‘5V < 5 ^ s V |s«*> ) <(5, 015^ | S' 0 ^ 5, 5, ' )SCW)g(5pSf') (12) 3 3 A simplifying assumption, based on empirical data for the H and He particles, specifies that the space wave function of these light re action products corresponds to relative S-state motion. We then write L-t (13) and the spin overlap becomes £ * ' X . S z X 5 J ) - <S (?) < S p $ } so7' ]O-tMt )<S/< W d s'o^ (14) ' M-tr I < T 1 rp ' £<0-1 84 and jl™ JvJz. iiA, -2jAi uA^jf <L/\sr I j- C C S ' T ^ I s n o . w * l5V'> r S i < +£«*> 4;t«) T - _ X . ‘ A ' [ 9 j ] < j j M $ |L a > < L .A S I I i ^ > j1 * 0,01- 5 ,Si <-A 5 2 < 5 P(T f S V j J fcM t > X £ j3. £ y s'cr' < 4C cf } 1 V i 3+' \ 4 3 | 4 * ; c 4>™ ((i P)) (15) With these expansions we are now in a position to construct an ex pression for the differential cross-section. For unpolarized projec tiles and unpolarized targets, this expression is related to the transition amplitude by where S indicates a summation over final states and an average over initial states: Ka and ya , and and y^ are the wave vectors and reduced mass of the incident probe and the light reaction product, 85 respectively. ^ -----! ------ y i T f- d s i c z r r k 1) 2 b 3 rzJ^OCUp+O (17) <pMt ^ t b ^ J*. | J . M i X ^ i j'*' l*CM C > * ,<L)iaf|j1 1 .><1.w * - | 3 W > - 5£ 5 £ # 5p(Tp s£ | J ^ h ^ . ' X 5 p(Tp S1 ^ ' j S ± ^ __ (18) ^ sX loj]* xl’KlL*y < ' < ' v I W „ l < ’ f ; , 4 ' ’> ! 1 . £ , £i-> i ' Recognizing that r: <cTfMj j“-1 5; j ' u , ' i c r t 'Mt -> = C s M i ' f t , - 'Zj+'l 0 0 ^_1 52 ] Jt^e^’ CSpOp 5*2^' | = — ! : — £ X , < r P ^ 25 + • i z z 2. <<-A5sIj^Xi.Vss );^> = ^ s y s , ’ Z.L-t' i AA we find <s(<r _ 2 J (z-rrff UA (ZS+t)(ZLr,) ( j 5 J iJ i -€(^ , l 5,5 : tt) | v;^ ^ | +>,><*>'* t*.) 4 " ’ <m > (19) where we have set Sp = 1/2 and Jt = 1/2. 86 A simplification of the integrals implied in the evaluation of the matrix elements can be realized by transforming the wave functions where n£A , NL'A' are quantum numbers describing the relative and center-of-mass motions of the transferred pair. Explicit expressions for the transformation coefficients can be obtained only for harmonic oscillator wave functions and these have been tabulated by Brody & Moshinsky [31]. In this case, the summation is restricted according to where n and N are harmonic oscillator quantum numbers. Note that the relative S-state motion assumed in the incident particle overlaps only with the X = 0 terms. We, consequently, choose the single particle wave functions to be normalized oscillator functions with length parameter v ; <()^j (R) and <|>^2 (R) into functions of relative and center-of- mass coordinates, A I A 2 , ^ < * A , N L-y u I L>fiAL'A'|LA)^(( o)4^L(R) (20) 1 4 / ( ? bJL* X A ' 2(n^ - + • 112) + £ + ^ = 2 (n + N) + ^ +A (21) (22a) (22b) 87 To perform the integrations over the internal coordinates of the triton and p , a gaussian form is chosen, that is: V ( V & e . & p (-/3Z/2;k ) (23) (24) where VD is the strength and g-1 the range of the potential; N is the normalization constant and n a length parameter for the wave function. The sum is over all pairs of nucleons in the triton and yields 3 2? 2 L , , V'Cftjit) = 2.^. 0 = 1 w L - 3 4 °^)= <k'°c4'>en}) (25) Here < j > 10 (vX2) is a normalized oscillator wave function: % [ tt] e-Kp (26) Inserting these expressions into the differential cross-section yields if _ k b y ' ( v a M ' ) ' LA J S C Z 5 + l ) f Z L + - | ; i s u ^ ' x * * a‘ S. S i M N * < MOj MLj l) ia,/, ( 4 * f It1 ) * (27) j caip L - ( S l n } ) | i l T i p ) U (■ (?) X. 4 £p 88 Rearranging dr = / > * / < , , J-' — ! _ r . ' A ' M ^ ' f i'L f ^ ( z n ^ r T k Z 5 + ' mm JS * ^ R £>, WL- l j * , A « i l x L > * ( 28> * J 4 ‘* * * 4 ' aCdi,Ul ) 1 1 ^ * 4 % ( Z p ) V 7 e ) % Mc e ) J e X r In this form, we have explicitly combined related terms. For example, the relative motion which, in the nucleus is described by < } > 10(r) , has to overlap with the motion of the pair in the light nuclide to which they are transferred. This overlap is denoted by -rt-n ~ / U " ° (/ > ) 4 ^ ) f > Z = C ' 1 /z Ci-y) I (2 9 ) "h“ e ^ ( - ) ' 1 x = , ~ 7 7 ~ , > y ^ * The last integral, K l~ 2 ^ CzLr,) ^ (4% >tlL^ 2 ^ 6 fi 4 C * t t y ) V NCI?) '/" ( & ) JlKJitf, (3 0 ) represents the probability amplitude for transferring a structureless nuclide from the orbital state N, L in a structureless nucleus. If we further define ^ M C Ss * H L ) 9 (3 1 ) y 89 we have concentrated as much of the nuclear structure information in G as is possible: A' is the parentage factor connecting the nucleus (A+2) to (A) and measures the degree to which the ground state of (A+2) has as its parent the state of the nucleus (A) that is formed in the interaction plus two nucleons in the state 6 = n2&2 ••LiSiJiT , is the overlap of the relative motions, and | ^ is the amplitude of the relative S-state motion in the bound state of the pair. This factor tends to favor single-particle states for which L = 0 or £1+ . These definitions thus result in a complete separation of the nuclear- structure calculation from the calculation of the transfer amplitude. Although Equation (28) is general in its application, its de velopment has required that we distinguish between the nucleons in the incident particle and those in the target. The effect of antisymmetri- zing the total wave function is to introduce exchange integrals. How ever , the exchange integrals, because they involve the overlap between bound and free states, will be smaller than the direct term [1, 2]- The evaluation of such effects has been done in the plane-wave approxi mation and verifies this assertion. We thus ignore these integrals in this development. However, in the reduction of the integral to the form of Equation (28), the overlap between Xa+2 an^ will contribute 90 a factor C - (T-jM-l , TMT|TfMTf). With this correction for antisymmetrization of the wave function, the expression for the differential cross-section becomes 3 T A A ^ ^ y i ’ ZSg'1 '1 L.3JM l Z3+< ) N (32) where we have also generalized the expression to include other two nucleon transfer reactions by introducing a parameter b t whose value depends on the reaction under consideration. Specifically: yS T ~ $T, 1 + Sjjo < * T , i ( y | > (Ss,c> STli ~ ^ 5 , 1 ^T,o^ + + ^ Tc> ^ V ~ ^s, i St-i0 for (p,t) for (n, ^He) for (p,^He) for (n,t) for (d,a) This expression requires an incoherent sum over L,S,J of the pair of transferred nucleons. However, the several radial states which enter into the interaction contribute coherently to the cross-section. The relative weights with which they contribute are determined by the structure factors G. For single-nucleon stripping reactions, only one principal quantum number contributes to the reaction and the Born approximation yields the following equation for the differential cross-section for (p,d) reactions: 91 Predictions based on this expression yield angular distribu tions of the same form as originally presented by Butler [32, 33]. To completely describe either of these direct interaction Due to the complexity of this expression, it is customary to make a zero range approximation which essentially reduces the calculation of the matrix elements to a single integral. For this approximation we set The implication of this assumption is that either the internal wave function of the outgoing projectile, or the interaction potential has a range short enough that it may be replaced by a delta function. Most computer codes indeed perform DWBA calculations under this simplifying assumption [1] . 19 3 Calculation of Theoretical Cross-Sections for the F(p, He) Reactions the unavailability of a suitable two nucleon transfer computer code, and the complexity of the calculations, it was necessary to separate the evaluation of the expression for the differential cross-section into two independent calculations. The first of these determined the Based on the derivation of the previous section, we can now 19 3 calculate the theoretical angular distributions for the F(p, He) and 19 F(p,t) ground and first excited state reactions. However, due to 92 structure factors, , assuming the ^ F target nucleus consisted of two nucleons and one proton outside an inert ^0 ground state core 19 and using the F shell model wave function of Elliott and Flowers [47] to determine the parentage of this nucleus with the ground and first excited states of the residual ^F and ^0 nuclei. The corresponding LM reaction amplitudes, , were estimated from angular distributions predicted by a single nucleon transfer computer code for the pick-up of a structureless "deuteron" or "di-neutron" cluster. Cluster Transfer Calculations LM The reaction amplitudes, B^gj , were determined from the angular distributions predicted by the single nucleon DWBA computer code JULIE [34] developed at the Oak Ridge National Laboratories. The ver sion of the code available predicted angular distributions in the zero- range approximation using an expression similar to Equation (33) . For 19 3 the F(p, He) transitions the calculations assumed the transfer of a "heavy" nucleon of charge 1, mass 2, and spin 1. LM The wave functions required to evaluate the integral for B^g^ are (Rp) describing the elastic scattering of 46 MeV protons in the entrance channel, and » describing 30 - 36 MeV He particles in the exit channel. Both wave functions were obtained from nuclear optical model data available in the literature. This model is based on the success of the shell model which indicates that, due to the Pauli uncertainty principle, nuclei are partially transparent to low energy nucleons. Absorption processes 93 can, to a good approximation, be lumped together and be treated as a process that removes particles from the incident beam. The target nucleus thus acts on the incident nucleus like a potential well, and the possibility of forming a compound state due to a collision may be taken into account by allowing the potential well to be complex. Such a model cannot account for the features of nuclear reactions that de pend on the characteristic structure of the compound nucleus. Never theless, it has been found to account for the main variations of the elastic differential cross-sections and the total reaction cross- sections with atomic weight and energy of the incident nucleon. A large number of different analytical forms have been adopted for the radial variation of the optical potentials. Initially, they were chosen for analytical convenience (i.e., square wells) but, with the availability of digital computers, it became possible to use more realistic forms that took account of the diffuseness of the nuclear surface. All such potentials are characterized by a nuclear radius, R0 , and a parameter, a, giving a measure of the surface diffuseness. Consequently, the real or refracting potential, V(r) , is usually assumed to have a Woods-Saxon form V(r) = Vf(r) where f(r) = Woods-Saxon form factor = {exp [(r-r0A ^ )/a] + 1} ^ , and Rq = nuclear radius = r0A^^ fermi On the other hand, the imaginary part of the potential accounts for many possible reactions that may well occur preferentially in certain regions of the nucleus and there is not a priori reason to suppose it 94 will be uniform in the nuclear interior. In fact present evidence indicates the imaginary absorbing potential is peaked at the nuclear surface. Such a potential requires a third parameter to specify it completely. As a result of this uncertainty, the imaginary potential, W(r), is usually comprised of a volume, Wv , and a surface term, Wg; W(r) = Wvf(r) + 4 Ws M i l l dr Although this potential assumes the Woods-Saxon form factor, there is no reason to suppose the real and imaginary potential should have the same radial variation. In fact, it has been found that if the radii of the two potentials are assumed to be the same, then the absorption must be peaked at the nuclear surface to give the observed proton reaction cross-section. However, if the imaginary part is permitted to extend beyond the real part, the observed reaction cross-section can be obtained with a volume absorption alone. An optical potential composed of a real and imaginary part of this nature has been found to predict the major characteristics of the elastic scattering differential cross-sections to the extent, that four (4) semi-empirical rules have evolved: 1) Overall scattering is increased and the diffraction pattern is shifted towards smaller angles when V or r is increased, 2) The positions of the maxima and minima are determined O mainly by the value of V r , 3) The amplitude of the diffraction pattern is damped when the absorbing potential W is increased, and, 4) Increasing the surface diffuseness parameters attenuates (MeV) 40 -20 > 10 - 1000 100 NUCLEON ENERGY ENERGY DEPENDENCE OF REAL AND IMAGINARY OPTICAL MODEL POTENTIALS FIGURE 24 96 the diffraction pattern, and, to a lesser extent, shifts it towards smaller angles. This rotates the diffraction pattern in a clockwise direction. Both the real and imaginary potential depths show considerable scatter about an overall energy variation when plotted as a function of the incident nucleon energy as in Figure 24. This is due partly to experi mental errors in the measured cross-sections and partly to ambiquities in the fitting procedure; the most notable of which is the V r11 ambiquity. Provided VrQ is kept constant rQ can often be varied in the range 1.2f to 1.35f without substantially affecting the fit to the experimental data. The overall energy variation of the parameters can be partly understood as due to the non-locality of the interaction. The inci dent particle is extended in space and so the potential it experiences is not simply a function of its vector distance from the scattering center. If this is taken into account in the calculations, the energy variation of the real part of the optical potential can be accounted for. However, since the use of local potentials significantly re duces the numerical complexity of the optical model calculations and since it has been shown [35] that predictions based on a non-local potential do not destroy the fits obtained by the local potential, most optical model fitting procedures employ a local potential. The gradual increase of the absorbing potential, W , with energy may be understood as a result of the Pauli principle. This forbids collisions that would send a particle into a state that is 97 already occupied, and thus inhibits absorption inside the nucleus and gives a resultant surface peaking. Since the states of low energy are filled, this effect is more restrictive at low incident energies and becomes progressively less important at higher energies. There are no apparent systematic variations of the radius parameter, r0 , or surface diffuseness parameter, a , either with energy or atomic weight. On the basis of experimental data which show that the scattered particles are generally polarized, one additional contribution must be included to completely describe the optical potential. This is the inclusion of a spin-orbit term. Since symmetry requires the spin orbit forces to be zero in the nuclear interior, the spin dependent potential may be expected to be small inside the nucleus and appreci able only in the surface region. Again, independent radial parameters are frequently used. To summarize then, the optical model analysis usually assumes a general interaction potential of the form UUI)- - V l W ' f ' - i i r * (A i i ) ' where „ x = ciz-i^A *)/*., ? (n - n l a 1/3) / aJ to determine parameters which describe experimental elastic scattering data as a function of incident energy and atomic number. 98 19 For the proton elastic scattering from F no such theoretical fits to experimental data have been made. This is due both to the 19 fact that F is a gas and a reactive target as well as to the inabili ty to experimentally separate the elastic and inelastic energy levels. Previous DWBA analyses [6, 7, 36, 37] at 40-50 MeV incident proton 1 9 energy used either interpolated optical model parameters from C , 016 , Al27 , or Si^® parameters reported in the literature or used elastic scattering parameters for protons elastically scattered from "^0. A third alternative would be to use the average optical model parameters determined for a specific range of incident energy on a specific range of nuclei. These average optical model parameters are available for incident proton energies in the range 9.5 to 22.2 MeV [38, 39, 40] or 30-40 MeV [41, 42] or for the 1-P shell nuclei [43]. Thus, due to the large variance in excitation energy and due to the shell structure of the various nuclei involved, this method might pro vide optical model parameters of questionable validity for the present experiment. An earlier analysis [36] of a scattering experiment with 30 MeV 19 protons incident in a F target used proton parameters obtained from an analysis of 31 MeV proton elastic scattering from ^0. Several other sets of proton parameters which equally well produced the 31 MeV 1 fi P+ 0 data were tried in this analysis as were three sets of proton parameters used in previous experiments by different authors. The sets of proton parameters which equally well represented the 31 MeV elastic proton data yielded very similar theoretical angular distri butions . 99 Due to the difference in the bombarding energies, it was decided in the present experiment to interpolate proton parameters 12 16 27 from optical model parameters in the literature for C, 0, Al, 2 8 and Si obtained from 40 MeV proton elastic scattering experiments [41, 42]. The results of this interpolation yielded parameters in agreement with those used by Hird & Li [37] in their analysis of the 19 F(p,a) reaction at 45 MeV. Figure 25 shows the ratio to Rutherford elastic scattering cross-sections used for the interpolation of the 19 F(p,p) parameters. Also, shown are the results of the interpolation. After several preliminary theoretical angular distributions had been 90 obtained, the results of a Ne(p,p) experiment became available to the writer [44]. The elastic scattering cross-section for this data is also shown in Figure 25 for comparison with the interpolated para meters used earlier. Although the ratio to Rutherford angular distri bution obtained from the interpolate optical model parameters agrees with the overall trend * subsequent theoretical prediction utilyzed 90 the Ne parameters for the incident channel. In contrast, optical model parameters describing the wave 3 19 3 function of the He particle in the exit channel of the F(p, He) reaction could ostensibly be obtained from theoretical fits to experi- 17 3 3 17 mental data of the 0(.He, He) experiment since 0 is a stable iso tope. However, the experiment has not been performed to this writers knowledge. This absence of experimental data requires that optical 3 model parameters describing 30 to 35 MeV He elastic scattering in the exit channel be obtained from a similar reaction on neighboring nuclei. & 0> re Ui ►- a p Q sa cc 40 60 8 0 100 120 ®CM RATIO TO RUTHERFORD ELASTIC SCATTERING CROSS- SECTIONS FOR NUCLEI BETWEEN A = 12 AND A = 28 FIGURE 25 101 16 A A Consider then the 0 ( He, He) reaction performed at 45 MeV [45]. In this instance, we have tightly bound particles of spin zero 17 3 3 incident on a doubly magic target. For the 0 ( He, He) reaction, however, we have a loosely bound particle of spin 1/2 incident on a target of spin 1/2. Thus, the parameters describing this experiment were deemed to be a poor representation of the desired elastic scat tering wave function. 1Q o q Data [46] is present in the literature for the F( He, He) reaction which meets most of the requirements of an alternate experi ment. However, this data was obtained using a CF2 (Teflon) target and 12 the final angular distribution was obtained by subtracting the C (^He, %e) contribution utilyzing data from a second experiment. Since this procedure introduces significant error, it was decided to use instead, optical model parameters which provided a reasonable fit to the 28.9 MeV %e + ^0 elastic scattering data [46]. Theoretical fits to this data were obtained by Dickens, et al [36] and no further optical model analyses were performed or considered for these exit waves. The optical model parameters used in the analysis of the experi mental data are shown in Table 12. The value of the real well depth 3 16 of the He + 0 elastic scattering optical potential is not the small est value obtained in the analysis of Dickens, et al. Rather it is one of a "family" of depths which yield good fits to the experimental data. This large value of the real well depth results in each com puted partial wave comprising the distorted wave having several 102 oscillations inside the nuclear interior and thus the contribution to the transition amplitude from inside the nucleus is substantially re duced. The alternative for obtaining the transition amplitude would be to use the smallest value of the "family" and employ a cut-off radius to reduce the contribution from the nuclear interior. This method of using a large value for the real well depth effectively reduces the contribution from the interior in a smooth and plausible manner and avoids the need for justifying a particular value for the cut-off radius. TABLE 12 OPTICAL MODEL PARAMETERS DESCRIBING ELASTIC SCATTERING IN THE ENTRANCE AND EXIT CHANNELS V(MeV) rR(f) aR(f) 4WD(MeV) Wv(MeV) rx(f) ax(f) rc(f) 20Ne (p,p) 43 1.14 0.70 21.7 9.9 1.34 0.66 1.25 160(3He,3He) 170 1.03 0.89 0.0 15.0 2.06 0.51 1.25 Using the optical model parameters of Table 12 to describe the entrance and exit channels and the quantum numbers of Tables 10 and 11 to describe the transferred cluster, a series of JULIE calculations were performed in which the value of each parameter was individually varied over a reasonable range. The theoretical predictions proved to be, to a large degree, insensitive to small variations of the entrance channel optical parameters. Similar variation of the parameters de scribing the exit channel, however, produced curves bearing little 103 resemblance to the experimental data. This was also the case when a cut-off radius was introduced and varied independently over a wide range. For the analysis, the optical potentials were as shown in Table 12 and no cut-off radius was employed. The last parameters to be determined are those which describe the potential well which binds the transferred deuteron cluster to the 170 core. The dependence of the ^F(p,^He) ^70 and ^F(p,^He) ^70 angular distributions on the choice of radius and diffuseness para meters for this potential is shown in Figures 26 and 27, respectively. In these calculations the cluster binding energy was given the experi mental value and the well depth was adjusted to obtain an energy level with the proper quantum numbers at this value. The predictions for the ground state angular distribution show that calculations using a large radius (rQ ~ 1.9F) for the extent of the binding potential yields a better fit to the data at small angles. On the other hand, calcu lations for the 0.871 MeV state showed better agreement with a small radius (r0 ~ 1.0F). In fact, in the latter case, an angular shift toward smaller angles by 5° would yield an almost perfect fit to the data. Such a shift would also improve the prediction for the ground state reaction. Attempts were made to shift the calculated angular distributions toward smaller angles by varying the remaining 'loose' parameters. These are the diffuseness and radius parameters of the potential well binding the transferred cluster. The attempts were, however, unsuc cessful. The conclusion for the present predictions must be that DIFFERENTIAL CROSS SECTION (jJS/STER) F ( P , He) 0 G R O U N D STATE 1000 100 1.0 v C mev) 57 134 R (F) 1.90 1.00 0.1 1 0 0 60 20 60 © C M THEORETICAL DIFFERENTIAL CROSS-SECTIONS FOR THE - 19F(p,3He) REACTION TO THE GROUND STATE OF 170 FIGURE 26 DIFFERENTIAL CROSS SECTION 0-10/STER) 105 0 . 6 7 1 M E V S T A T E 1000 100 R(F) V(MEV) 1.90 36 1.00 76 0 . 1 100 20 60 60 © C M | THEORETICAL DIFFERENTIAL CROSS-SECTION FOR THE l9F(p,3He) REACTION TO THE 0.871 MEV STATE OF 170 FIGURE 27 106 these radii are required to obtain theoretical distributions with approximately the correct shape. Having established optical model parameters and bound state well parameters consistent with the experimental data, we are now in a position to determine the reaction amplitudes. The specific amplitudes, LM ®NSJ 5 to determined are defined by the summations of equation (32) over the LSJM quantum numbers given in Tables 10 and 11. Thus, in the evaluation of the expression there will be, for the ground state transition, five terms in the M summation, and, for each value of S_, there will be two contributing values of J_. For the first excited state transition, the angular momentum transfer is zero and the summation over angular momentum sub states reduces to a single term. There is also a simplification in the summation since, for L = 0, . J = S_ and we have a single value L M of J. 17 Consider first the reaction to the ground state of 0. Expand ing the summations over LSJ in Equation (32) , we find NZ5J1 N l t > 7 i (4*) \ a s x K n d rts-Z. l — ’ MZlZo W»2 N N z n Aj i 3 ' 1 107 + Hi- IS <S) S> U Z o Z i U & 2 . AJ e>2/1 2.°3 l N&3 However, the transition for which L = 2, S = 0, J = 3 is not permitted by the angular momentum selection rules and we then have (34) 2M 2M 2M The reaction amplitudes for B^^ > %13 anc* BN02 must thus be deter mined for values of N consistent with the limitations imposed by Equation (21) . Figure 28 shows the relative magnitudes of the angular distributions, OnlsJ o t [ JI » predicted by the JULIE code for these quantum numbers. Each of these angular distributions were computed using the experimental binding energy. Consequently, the well depth of the binding potential for N = 1 was considerably smaller than that obtained for N = 2. The alternate method would have been to use the deeper potential obtained for N = 1. This would have yielded even smaller cross-sections for 0^202 tBian shown in the Figure. Referring to Figure 28, one sees that the computed angular distributions for 0£212 anc* a2213 were very similar in shape and magnitude to that for 02202 anc* we assume B2202 = b2212 “ B2213 (35> and b1201 = b1212 “ b1213 (36) 108: ! I i GROUND S TATE 22\3 £ Z D V -> w J z b 60 40 100 20 3 RELATIVE DIFFERENTIAL CROSS-SECTIONS FOR DETERMINATION OF REACTION AMPLITUDES FOR 19F(p,3He) REACTION TO GROUND STATE OF 170 FIGURE 28 f 109 One also notices that the distributions for the two contributing values of the principal quantum number, N, are very similar in shape, but differ in magnitude by a factor of four or more over most of the angular range. Thus, it seems reasonable to assume that IB2202I ~ 2 Ib1202 t• To fix the sign of the amplitudes, we note that, due to the choice of 3 the internal He potential, most of the contribution to I%202 should come from r >_ r (nuclear) and since U3"® (R)/U^(R) is negative in this region, we assume B2202 ~ “ 2B1202 ’ LM The determination of the B contributing to the 0.871 MeV NSJ & excited state transition is less complicated since the angular momentum transfer is zero, and the summation over angular momentum substates reduces to a single term. There is also a simplification in the J summation, since for L = 0, J = S and we have a single value of J. The expression for ( ) then reduces to . z S> I f > J o&c>\ fOOo J # N (38) Figure 29 shows the relative magnitudes of the cross-sections for these quantum numbers and the contributing values of N. Making assumptions similar to those made for the ground state transition, we find * i p >' 6 ° ° | + j d-a _ Z-1 Lj U O M o /Ul| I (o - e.% M (ARBITRARY UNITS) 0.871 M C V S T A T E O ' 3000 -> ’ton 20 100 4 0 60 8 0 CM . RELATIVE DIFFERENTIAL CROSS-SECTIONS FOR DETERMINATION OF REACTION AMPLITUDES FOR 19F(p,3He) REACTION TO 0.871 MEV STATE OF 170 FIGURE 29 Structure Factor Calculations LM The structure factors, %Lgjx » corresponding to the B^Sj calculated for the ground and first excited state reactions are de fined by Equation (30) to be y where the NLSJT values are specified by Equations (34) and (38), respectively. The coefficients of fractional parentage were calculated using 3 the Elliot and Flowers wave function [47] which is 12% d (i.e., all 2 3 three extra-core nucleons in the Id shell), 59% d s, and 29% s . The total wave function is a linear combination of wave functions iKdn s3 n) , ^(19F) = -0.35 iKd3) - 0.77 i p ( d 2 s ) + 0.54 ^(s3) where that portion of the total wave function that corresponds to in completely closed Is and lp orbitals has been neglected. Since this assumption permits only the description of states of positive parity [48], our calculations are restricted to the ground and first excited state transitions. 112 19 The coefficients of fractional parentage between this F ground state configuration and the ground and first excited states of the "^0 nucleus have been calculated [36] assuming these states to consist of 16 1 A an inert 0 core plus one nucleon in th W 5 / 2 shell and an 0 core plus one nucleon in the ^ 1/2 orbital, respectively. These coefficients were used in the present calculations. To evaluate the overlap jfln, the oscillator parameter, v , was -1/3 -2 given a value of A f which corresponds to the oscillator spacing h u 8 41 A MeV used by Elliott and Flowers [47]. The size para meter p of the A = 3 nuclide is connected to its mean-square radius by n2 = 1/6 <R2> . The experimental RMS radius of 1.97 F was used and yielded a size parameter of 0.207 F-^ [5]. The transformation coefficients were obtained from the tabula tion of Brody and Moshinsky [31] subject to the limitation imposed by Equation (21). In this manner, the structure factors for the ground state were found to be: g22120 = -0.016 g22130 = g22120 g12120 = - 0.011 g12130 = g12120 g22021 = G-22120 G12021 = -g12120 and for the transition to the 0.871 MeV state: 113 g30001 = -0.122 G20001 = -0.011 G10001 = -0.005 g30110 -G30001 G20110 -G20001 G10110 "G10001 Cross-Section Calculations: With both the reaction amplitudes and the structure factors da calculated we are now able to evaluate Equation (34) for z do-' d - a Mi-? ^ Z * & ^ U l \ T g &■ 1-bJ- W 2ll o W it I Z | t j AJl3 I d£ s"' „ -ZM TT / J Z 0 2 . I MO 2 Expanding: ■Z.M it . & + l 2 o I Z l 2 \ + 6„ J +<^se„„l62„l J'H 4 i k ,« . l C Iz* Z& 6 sz" 6 M * 1 e T f ' I ? 1 2 lt»2p -2i»2p (o 2 t"! 22021 * -22»2 1 J j or, da ’ i& m r £ < - 2 go</ h. , . z C a<2 I L- 4 1<£* Z z z \ i o 2« ^(2120 S ’ ppZ ^ 132,^ J C ^ Z Z l Z c &ZH I f I - f 7 ^12.130 ^ ‘ '3 2(3 ) *- ~ ----------- c Z H Z 2 ) 21 , 3 0 + -h 1 . 2 r s . J L I 2-o 2 I L . i. 6 'Z.Zc’ Z I 4 - 2 •2.M ^ I Z o Z I S ’i c . 2 ZS> 22 021 S ’ 2 .0 e 114 where we have dropped terms containing the ratios ( o . c . n ' ) (<=> .S) = O . O S & v * = 7 >a iz<=> f i . aM ^ I I Z ^■2.2. t 2 0 211 1 ^ 12.13 to 5 ^ 113 Z.HI& ft*” ■ L ^ U o U 2M 2 cz.ZM iio z (o.cn (o.s) - o . oSfo Substituting numerical values one obtains clo' e>2” Z Z o Z i 1 2.02 H--Z 1 2.2^2/ M--Z B Zzoii 2oZ 2 r- ( l . l & l ) | _ I + 2 (o. 6.T) C-o.S ) Finally, for the ground state transition 2 - . 2M 1 z Similarly, Equation (38) may be expanded to yield: < = " i c M c . TS >3 ,1 ^2ol I® ^2.1 1 ZS1301I O &311 + £ <s , C 2 “3 2opo 1 2 °, ° < 0 3 0 0 0 1 e>3o<; 3 2,0001 ,--7 2-ci f > 3 0 O or, finally •£o-l 2. . e * o . ( c \ dS? 5 ° ° 115 Summarizing, we have manipulated the expression for the two- nucleon transfer differential cross-section into a form which permits the structure factors, G, to be factored out of the angular momentum summations; i.e. A- f - 6 z s i r. d -Cl m We must now compare these theoretical predictions with the experimental data. One method is to obtain a theoretical angular dis- tribution atheo by setting G = 1.00 and determining a spectro scopic factor from the empirical data; The calculated structure factor, G, should then compare favorably with the spectroscopic factor. This method is particularly suited for the present calculations, since the JULIE computer code predicts o^eo directly. For greater reliability in these comparisons, we shall follow the recommendation of Towner and Hardy [1] and compare the ratio of the spectroscopic factor for the excited state to that of the ground state. Considering first the theoretical predictions, the ratio of the structure factors is: S Pexp atheo { A < r / A j x ) ^ 0 .0 I0 2 . . while the ratio of the spectroscopic factors is: ( < T gS / fog )e*p _ C / O V ) fUe& o , o i l . 115 Summarizing, we have manipulated the expression for the two- nucleon transfer differential cross-section into a form which permits the structure factors, G, to be factored out of the angular momentum summations; i.e. ~ c ^ 2 l l I l \ S--CL M We must now compare these theoretical predictions with the experimental data. One method is to obtain a theoretical angular dis- 2 tribution ^theo setting G =1.00 and determining a spectro scopic factor from the empirical data; S = atheo The calculated structure factor, G, should then compare favorably with the spectroscopic factor. This method is particularly suited for the present calculations, since the JULIE computer code predicts ot^eo directly. For greater reliability in these comparisons, we shall follow the recommendation of Towner and Hardy [1] and compare the ratio of the spectroscopic factor for the excited state to that of the ground state. Considering first the theoretical predictions, the ratio of the structure factors is: a ( <i<r/j-a.)6 » A / <Ttic*z o.VZ>___________________ ( d<r/ J XL ) / < r3o,, © . I 6 }o llo while the ratio of the spectroscopic factors is: ( < p g s / fey. ) e g p ^ o . o \ Z . . ( ^ t k e e , 116 Thus, the empirical ratio agrees very well with the ratio of the structure factors. 19 Calculation of the Theoretical Cross-Sections for the F(p,t) Reactions Cluster Transfer Calculations 19 3 19 In anology to the F(p, He) calculations, the F(p,t) transi tions assumed the transfer of a "heavy" nucleon of charge 0, mass 2, and spin 0. The same wave function was used to describe the entrance 19 3 channel as for the F(p> He) experiment. Optical model parameters describing the exit channel were more difficult to obtain. This is due not only to the absence of a suitable target but also due to the sparsity of neighboring nuclei which have been studied with triton probes of appropriate energy. However, a triton probe differs only 3 slightly from a He probe in mass, spin, and pairing energy. The principle difference is the isospin of the two probes. This difference of charge would be evidenced primarily in the Coulomb repulsion term of the optical potential and would not significantly effect the nuclear terms. Thus, the optical model parameters obtained 19 3 3 for the F( He, He) elastic scattering were also used to describe the 19 exit channel of the F(p,t) reaction. 19 3 A parameter study similar to that described for the F(p, He) cluster transfer was performed. In this reaction also, no dependence on the Coulomb well radius was found and it was maintained at a 1/3 constant value of 1.25 A F. The predictions for both the ground and 0.500 MeV state are shown in Figures 30 and 31. DIFFERENTIAL C R O S S S E C T IO N (JJB/STER) 117 I O O O O f( p,t ) ' f GROUND STATE 1000 100 10 20 40 SO 80 100 120 ©CM THEORETICAL DIFFERENTIAL CROSS-SECTION FOR THE 19F(p,t) REACTION TO THE GROUND STATE OF 17F FIGURE 30 DIFFERENTIAL C R O S S S E C TIO N (UB/STER) 118 I O O O O ,9;f (p,t) i7f 0 .5 0 0 MEV STATE 1000 100 20 40 SO 80 100 120 OcM - THEORETICAL DIFFERENTIAL CROSS-SECTION FOR THE 19F(p,t) REACTION TO THE 0.500 MEV STATE OF 17F FIGURE 31 119 Cross-Section Calculations The evaluation of the expression a sp 'ST ^--1 dr I __ |_M E c l ; 1^ , -- 6 L.STT M M WLST WSJ 19 for the F(p,t) reactions is most conveniently accomplished by determining that portion of da/dA which corresponds to an S = 0, T = 1 3 transfer for the (p, He) calculations. These are the only contributing terms since the selection rules prohibit the transfer of an S : 1, T = 0 cluster. Then for the ground state transition we find ■ I • z . «* 7 - 1 J- J-fL. T d b - z& *2 M N e> KJ2-OZI W O l and for the first excited state d<7 -d-rt * r AJ Nooo I Ajoo I • Evaluating these expressions under the same assumptions as 3 presented for the (p, He) reactions, we find L d - n . J ■z ri Z,o z ev Using these values, the ratio of the spectroscopic factors for the ground and first excited states is expected to be ( do" / dui )gnj j C T j C<dcr/d-a ) e * j < T : z z o i 3 COO 120 Determination of the ratio of the experimental and theoretical differential cross-sections yielded (agrd/aex^0„_ _exp. a 0.004. ^ gr<j/aex) theo Thus, the empirical and experimental ratios compare favorably. CHAPTER V SUMMATION 19 3 Experimental angular distributions for the F(p, He) and 19 17 F(p,t) reactions leading, to the ground and low-lying levels of 0 and -^F, respectively, have been obtained using incident protons of 46 MeV. The variation of these data with scattering angle indicate that both reactions proceed predominately by direct interaction processes. 19 3 The angular distributions for the F(p, He) 0 transitions to 1 7 the ground and 0.871, 3.06, and 3.85 MeV levels of 0 are consistent with the angular momentum transfers allowed by the selection rules. Similar conclusions can be drawn from the ^F(p,t) angular distri butions to the ground and 0.500 and 3.10 MeV levels of ^F. Theoretical predictions based on the transfer of a heavy cluster yield angular distributions which agree in overall structure with the experimental data but not in absolute magnitude. These calculations utilyzed optical model parameters from independent investigations pre sented in the literature to describe the elastic scattering in the entrance and exit channels. The radius and diffuseness parameters of the potential well which binds the transferred cluster to the re sidual nucleus were varied over a reasonable range to obtain the best fits to the data. In these calculations the cluster binding energy 121 122 was given the experimental value and the well depth was adjusted to obtain an energy level with the proper quantum numbers at this value. For the L-0 angular momentum transition to the 0.500 MeV state of ^F, these predictions indicate that the transfer of a heavy cluster is adeuqate to describe the experimental data. For reactions, such as the ^^F(p,^He) transition to the 0.871 MeV level of ^0, in which both £=0, and Z=2, transfers are permitted, this method predicts an angular distribution which is displaced from the experimental curve by about 5°. Efforts to remove this displacement by adjustments of the binding well parameters were unsuccessful. Results for the ground state L=2 19 1 transition in the F(p, He) reaction are not as encouraging in that the predicted inflection points are not as pronounced. Again, however, 19 the results for the equivalent F(p,t) reaction to the ground state of show better agreement with theory. Theoretical structure factors were calculated for the pick-up 19 of two nucleons from the d$/2 si/2 s^ells of F. A comparison of the ratio of these factors for the ground and first excited state with the experimental data for each reaction indicates that the phases of the Elliott and Flowers shell model wave function are consistent with the experimental results. These comparisons also show that the normalization of the ground state and the first excited state angular distirbutions are the same. No theoretical predictions were attempted 17 17 for the higher excited levels in 0 or F since these transitions require a change in parity and the configuration of the nucleons in these states is not certain. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. REFERENCES I. S. Towner and J. C. Hardy, Advan. Phys. 18^ 401 (1969) N. K. Glendenning, A. Rev. Nucl. Sci. 1^3, 191 (1963) W. Tobocman, Theory of Direct Nuclear Reactions. (Oxford Univer sity Press, 1961) G. R. Satchler, Nucl. Phys. 5 5 _ , 1 (1964) N. K. Glendenning, Phys. Rev. 137B, 102 (1965) J. J. Wesolowski, L. F. Hansen, J. G. Vidal, and M. L. Stelts, Phys. Rev. 148 , 1063 (1966) H. D. Holmgren and C. B. Fulmer, Phys. Rev. 132, 2644 (1963) K. A. Huber, Annual Progress Report of the Nuclear Physics Laboratory, University of Southern California, Los Angeles, California. 1967 (Unpublished) G. M. Temmer, Phys. Rev. Letters 1 2 _ , 330 (1964) R. K. Cole, R. Dittman, M. S. Sandhy, C. N. Waddell, and J. K. Dickens, Nucl. Phys. 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Scott, Annual Progress Report of the Linear Accelerator Laboratory, University of Minnesota, Minneapolis, Minnesota, 1966 (Unpublished) 25. G. M. Reynolds, J. R. Maxwell, and N. M. Hintz, Annual Progress Report of the Linear Accelerator Laboratory, University of Minne sota, Minneapolis, Minnesota, 1966. G. Reynolds, J. R. Maxwell, 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 125 N. M. Hintz, Bull. Amer. Phys. Soc. LO, 439 (1965) R. K. Cole, R. Dittman, Annual Progress Report of the Nuclear Physics Laboratory, University of Southern California, Los Angeles, California, 1967 (Unpublished) G. Bassani, N. M. Hintz, C. D. Kavaloski, J. R. Maxwell, and G. M. Reynolds, Phys. Rev. 139, B830 (1965) N. K. Glendenning, Nucl. Phys. 29^, 109 (1962) A. Messiah, Quantum Mechanics (North-Holland Publishing Company, Amsterdam, 1961) M. K. Banerjee, "Theory of Stripping and Pick-Up Reactions" in F. Ajzenberg Selove, Ed., Nuclear Spectroscopy (Academic Press, New York, 1960) T. A. Brody and M. Moshinsky, Tables of Transformation Brackets (Monografias del Institutu de Fisica, Mexico, 1960) S. T. Butler, Proc. Rog. Soc. (London) A208, 559 (1951) S. T. Butler, Nuclear Stripping Reactions (John Wiley, New York, 1957) R. H. Bassel, R. M. Drisko, and G. R. Satchler, Oak Ridge National Laboratory report 0RNL-3240, 1962 (Unpublished) P. E. Hodgson, The Optical Model of Elastic Scattering (Oxford University Press, 1963) 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 126 R. K. Cole, R. Dittman, H. S. Sandhu, C. N. Waddell, and J. K. Dickens, Oak Ridge National Laboratory Report ORNL-TM-1585. B. Hird and T. Y. Li, Can. J. Phys. 46_, 1273 (1968) B. Buck, Phys. Rev. 130, 712 (1963) F. G. Perey, Phys. Rev. 131, 745 (1963) L. Rosen, J. G. Beery, A. S. Goldhaber, and E. H. Auerback, Ann. of Phys. 34_, 96 (1965) M. P. Fricke, E. E. Gross, B. J. Morton, and A. Zucker, Phys. Rev. 156, 1207 (1967) M. P. Fricke and G. R. Satchler, Phys. Rev. 139. 567 (1965) B. A. Watson, P.P. Singh, R. E. Segal, Phys. Rev. 182, 977 (1969) N. Hintz, private communication. B. G. Harvey, Phys. Rev. 146, 712 (1966) H. M. Sen Gupta, J. Rotblat, P. E. Hodgson, and J. B. A. England, Nucl. Phys. 38^, 361 (1962) J. P. Elliott and B. H. Flowers, Proc. Rog. Soc. (London) A229, 536 (1955) M. Redlich, Phys. Rev. 99^, 1427 (1956) APPENDIX A Tabulation of Experimental Data for The ^F(p,^He) Reactions to the Ground, 0.871, 3.06, and 3.85 MeV 17 Levels of 0 and for 1 Q The F(p,t) Reactions to the Ground, 0.500, 3.10, and 3.86 MeV Levels of ^F. 127 c UNIVERS ITY OF S O UT HER N C'U I F 0 M I A N U C L E A R P H V S I C S L A 8 0 R A T 0 R V o :' • ; s t a t i s t i c a l e v a l u a t io n of oata acc um ulated FOR THE F I? IP *H E 3 I REACTION o • ; 0 - EXCITED STATE I - EXCITED STATE | 10 ♦ 1) EXCITED STATE o THETA RUN SIGMA ERROR DELTA SIGMA ERROR OELTA | SIGMA ERROR DELTA COR M3. IMM3I <MH8> AVG CMNBI IHMBI AVG | M M 8 ) 1 MHB1 AVG 1 1 . oa 3036 6 8 3 .5 5 6 0 .8 8 1 .5 0 2 3 6 .1 9 2 0 .2 5 1 0 .1 9 j 8 8 7 .7 6 6 5 .6 2 8 .3 7 o 3036 68 7 .8 1 5 5 .6 8 2 .7 6 176 .32 2 6 .6 7 1 7 .6 8 1 8 6 6 .1 2 6 1 .5 6 1 5 .2 6 6 8 5 .0 5 2 .0 3 6 .1 0 19 6 .0 0 13.62 6 0 .2 7 ! 8 7 9 .3 7 j 1 1 .3 0 3 3 .8 9 o . I 3 * 84 3033 5 9 3 .3 5 * 6 6 . 0 5 * 5 7 . 9 3 * 5 8 . 1 9 * 1 3 .7 9 * 1 5 .3 6 * 1 ! 6 6 1 .5 6 * 6 6 . 1 6 * 5 0 .7 0 * 3037 5 1 7 .6 0 3 1 .6 7 7 .8 2 8 0 .6 7 1 1 .9 9 6 .9 2 ; 6 0 6 .0 7 3 3 .8 7 6 .7 7 o 6026 * 9 7 . 0 7 3 9 .7 7 2 8 .3 5 8 6 .6 6 1 5 .3 8 0 .9 1 ! 5 8 1 .5 3 6 2 .6 6 2 9 .3 1 6039 5 1 2 .6 8 37.06 12 .7 6 9 2 .5 9 16 .5 6 9 .0 6 6 0 5 .2 7 3 9 .0 0 5 .5 7 5 2 5 .6 2 1 8 .8 6 5 6 .5 7 8 3 .5 5 5 .3 2 1 5 .9 7 | 6 1 0 .8 6 15 .0 7 65.21 n ✓ 5 1 0 .5 0 * 5 . 9 6 * 1 7 . 8 9 * . 8 8 .6 3 * 2 . 1 5 * 6 . 6 6 * 1 5 9 9 .2 5 * 7 . 5 8 * 2 2 . 7 5 * f 9 1 1 6 .6 0 j 2020 6 1 8 .2 1 * 2 1 . 3 5 * 2 0 . 1 2 * 7 6 . 2 2 * 6 . 7 5 * 1 1 .1 3 * j 6 9 6 .6 3 * 2 2 . 6 0 * 3 0 . 8 2 * 1 2 0 2 1 6 1 7 .3 3 1 9 .3 6 1 9 .2 3 6 1 .7 7 5 .9 7 3 .3 1 6 7 9 .1 0 2 0 .2 6 1 5 .5 0 3023 6 5 8 .9 1 * 6 7 . 3 2 * 6 0 . 8 1 * 7 1 .1 1 * 1 7 .9 1 * 6 .0 3 * j 5 3 0 .0 2 * 5 0 . 5 9 * 6 6 . 6 2 * n : « 3026 3 7 3 .2 0 2 3 .6 6 2 6 .9 0 7 3 .5 6 9 .5 0 8 . 6 5 • 6 6 6 .7 6 2 5 .6 8 1 6 .8 7 / 3026 5 2 1 .1 3 * 5 5 . 7 7 * 1 2 3 .0 6 * 7 1 . 1 6 * 1 9 .0 5 * 6 . 0 5 * 5 9 2 .2 7 * 5 9 . 1 9 * 1 2 6 .6 7 * } 3028 3 6 3 .3 7 * 6 0 . 5 1 * 3 6 . 7 3 * 6 1 . 6 7 * 1 3 .2 6 * 2 3 .6 2 * j 6 0 5 .0 3 * 6 2 . 6 2 * 5 0 .5 7 * o : ! 6006 3 8 5 .3 0 2 3 . 70 1 2 .8 0 6 3 .1 6 8 .5 8 1 .9 5 | 6 6 0 .6 6 2 5 .2 1 1 5 .1 7 I 6010 3 9 6 .6 8 2 8 .1 3 3 .6 2 6 0 .9 2 10.18 6 .1 6 6 5 5 .6 0 2 9 .9 2 6 .2 0 • i 5029 * 3 8 2 .0 5 2 5 .6 7 t 6 .0 5 6 7 .7 3 9 .8 2 2 .6 6 j 6 6 9 .7 8 2 7 .6 9 1 3 .8 3 o . v \ i 6023 6 1 8 .1 6 . 2 6 .5 3 2 0 .0 7 5 9 .7 8 9 .0 6 5 .3 1 6 7 7 .9 6 2 8 .0 3 1 6 .3 6 r 6065 3 6 7 .5 9 * 2 0 . 6 0 * 3 0 . 5 1 * 6 2 .8 6 * 7 . 3 6 * 2 .2 5 * 6 3 0 .6 3 * 2 1 .6 8 * 3 3 . 1 8 * ; . i 3 9 8 .1 0 9 .1 5 2 7 .6 5 6 5 .0 9 2 .6 0 7 .2 0 6 6 3 .6 1 9 .9 0 2 9 .9 3 O ' ' • . ! 3 9 6 .5 3 * 8 . 0 0 * 2 6 . 0 0 * 6 3 . 8 7 * 1 . 9 5 * 5 . 8 6 * 6 6 1 .3 1 * 6 . 6 5 * 1 9 .3 5 * o 1 9 *3 6 - 2019 | 2 5 7 .6 3 1 6 .1 6 2 2 .2 3 7 6 .1 2 6 .3 0 2 .3 5 j 3 3 3 .7 5 1 5 .6 8 2 0 .2 6 f 2022 2 9 9 .0 2 1 6 .6 6 1 9 .1 7 7 7 .6 3 6 .8 2 3 .6 5 3 7 6 .6 5 1 5 .9 7 2 2 .6 5 ; * 3022 ' 275 .91 1 3 .9 7 3 .9 6 6 8 .2 5 5 .9 6 5 .5 3 3 6 6 .1 6 1 5 .1 8 9 . 06 O . 3029 ... 2 6 6 .5 6 1 9 .9 0 15 .2 9 7 7 .1 9 1 0 .0 8 3 .6 1 1 3 6 1 .7 5 2 2 .3 1 1 2 .2 6 1 3030 2 7 6 .3 0 1 8 .6 6 5 .5 5 6 6 .1 2 8 .6 6 7 .6 5 3 6 0 .6 2 2 0 .6 6 1 3 .5 7 i 5028 2 9 3 .0 9 2 0 .3 0 1 3 .2 6 6 8 .6 1 8 .9 6 5 .3 7 3 6 1 .5 0 2 2 .1 9 7 .5 0 o 6020 ; 3 0 5 .8 6 * 2 0 . 6 9 * 2 5 . 9 5 * 9 0 . 7 7 * 1 0 .3 3 * 1 6 .9 9 * . 3 9 6 .6 1 * 2 2 . 9 5 * 6 2 . 6 1 * ■ { 2 7 9 .8 5 6 .8 3 2 0 .6 8 7 3 .7 7 2 .7 0 6 .1 1 3 5 3 .9 9 8 . 2 2 2 6 .6 5 .. A ; 2 7 7 .1 3 * 6 . 7 9 * 2 0 . 3 8 * 7 2 . 3 6 * 2 . 0 6 * 6 . 1 7 * 3 6 9 .7 1 * 7 . 0 0 * 2 0 . 9 9 * O . .: i . ‘ ■ ; ! rs ■ ’; i i O ' ■ • ■ ■ o ' o o 0 !° Ic 1 !C i O' i ! o o- o i " i O o o o 0 o; o' o o U N I V E R S I T Y NUCLEAR OF S O U T H E R N CAL IF O R PHY SI CS L A B O R A T O R Y STATISTICAL EVALUATION OF OATA ACCUMULATED FOR THE FW IPuHESJ REACTION i- . 1 0 - EXCITED STATE I - EXCITED STATE ( 0 « 11 EXCITED STATE THFTA RUN sig m a ERROR DELTA SIGMA ERROR DELTA SIGMA ERROR DELTA COM N3. (MMB) 1HHR) AVG ( mmbi IMMfll AVG (MMftl (MMftl AVG 2 2 .1 2 2000 2 1 0 .2 2 6 1 2 .7 0 6 2 1 .7 7 6 9 0 .2 1 6 7 .7 1 * 6 .9 1 6 30 0 .6 3 6 16 .8 5 6 2 7 .5 5 6 20lft 19 7 .3 0 10 .5 6 3 .9 3 8 9 .7 6 6 .0 7 6 .6 6 2 8 7 .1 2 12 .1 8 1 6 .2 6 3000 19 0 .3 3 2 0 .3 0 0 .1 2 0 6 .7 0 13 .0 0 3 .6 8 2 6 7 .1 1 2 6 .5 5 5 .7 6 7007 IR 6 .0 5 1 0 .5 0 2 . 6 0 * 6 9 .3 6 5.B 9 1 3 .9 5 2 5 5 .3 9 12 .0 6 1 7 .6 9 3009 1 6 7 .5 3 6 1 2 .6 9 6 2 0 .9 2 6 7 6 .3 3 6 8 .2 0 6 6 .9 6 6 26 3 .8 6 6 1 5 .1 1 6 2 9 .0 1 6 • 4 3010 1 9 6 .9 2 1 3 .2 5 3 .6 7 7 3 .6 6 7 .6 6 9 .0 3 2 6 ft.39 1 5 .3 0 6 .6 9 » 3031 2 1 1 .'0 6 12 •6 6 6 2 2 .0 5 6 0 5 .3 6 6 7 .6 2 * 2 .0 6 6 2 96 .6 6 6 1 6 .6 8 6 2 3 .7 6 6 i 6001 I 0 5 .9 5 1 3 .0 0 2 .5 0 A9 .7 7 9 .1 7 6 .6 7 2 7 5 .7 2 1 6 .5 7 2 .8 6 r \ * 6012 16 7 .3 7 6 1 1 .0 9 6 3 1 .0 0 6 0 5 .2 9 6 ft.626 2 .0 0 6 2 6 2 .6 7 6 16 .5 0 6 3 0 .2 1 6 6027 19 6 .0 7 17.6ft 9 .6 2 0 5 .6 3 A. 28 2 .3 1 2 0 3 .6 7 1 5 .0 ? 1 0 .7 9 * 6031 1 6 0 .0 0 1 6 .0 0 3 .6 5 13 3 .1 6 1 0 .2 5 1 9 .0 6 2 0 3 .1 3 17 .3 5 1 0 .2 5 169 .65 5 .1 3 1 5 .7 9 0 3 .3 0 2 .6 5 ft. 56 2 7 2 .0 0 6 .1 6 1 8 .6 6 i 1 90 .1 7 6 2 .7 0 6 0 .3 6 6 0 2 .6 1 6 6 .3 1 6 12 .9 2 6 2 7 6 .0 0 6 6 .9 1 6 1 6 .7 6 6 - 2 6 .8 7 2001 1 5 6 .5 6 1 1 .9 5 6 .1 6 1 1 1 .5 2 9 .9 1 1 6 .2 9 2 6 7 .0 8 1 5 .5 5 6 . 7 0 ; • 3021 1 5 9 .6 0 1 1 .6 6 6 .2 9 0 3 .1 9 7 .7 7 1 5 .0 5 2 3 9 .5 9 1 3 .0 5 2 0 .7 9 i 5026 - 17 5 .6 5 12 .5 9 1 1 .7 5 9 2 .5 0 f t.73 2 .6 5 26A .03 1 5 .3 2 7 .6 6 i 6021 166 .02 1 2 .3 6 2 .3 3 106 .10 9.5ft 1 0 .8 6 2 7 2 .1 2 1 5 .6 6 1 1 .7 5 I 1 6 3 .7 0 6 .2 0 1 2 .6 6 9 5 .2 6 7 .1 7 2 1 .5 1 2 6 0 .3 7 7 . A3 2 3 .6 8 2 7 *6 2 - 2002 1 6 1 .2 6 7 .3 6 5 .6 2 92 .6 0 5 .6 3 9 . 5 6 2 3 3 .7 6 9 .1 3 1 5 .0 7 3020 1 3 5 .3 0 6.A2 0 .3 5 7 7 .5 6 6 .7 8 5 .3 3 2 1 2 .0 6 8 .3 3 5 .0 1 6011 16 3 .6 1 6 1 5 .1 2 6 2 7 .9 7 6 100 .936 11 .626 10.036 26 6 .5 6 6 1 9 .0 7 6 6 5 .0 7 6 i 5025 13 0 .1 9 10.63 5 .6 5 8 0 .0 6 . 7 .9 3 2 .8 6 2 1 0 .2 3 1 3 .1 0 8 .6 6 t 6038 1 2 0 .0 7 9 .6 0 15 .5 7 7 3 .3 2 7 .2 9 9 .5 0 1 9 3 .3 9 1 2 .0 5 2 5 .2 0 13 5 .6 6 5 .1 8 1 5 .5 3 8 2 .9 0 6 .2 6 1 2 .7 3 2 1 8 .6 7 9 .0 0 2 7 .2 6 . . * 1 3 3 .6 1 6 6 .2 6 6 1 2 .7 2 6 * 0 1 .7 1 6 6 .2 6 6 12.726 2 1 5 .6 5 6 8 .0 6 6 2 6 .1 7 6 1 0 .3 6 . 2003 1 3 3 .6 6 7 .5 7 5 .7 3 7 9 .6 6 5.6ft 5 .2 1 2 1 3 .1 0 9 .3 5 11.91 3016 12 0 .2 5 6 .0 6 5 .6 7 6 5 .5 2 5 .6 6 f t.72 1 8 5 .7 7 9 .8 3 1 5 .6 2 7019 126.61 5 .9 1 0 .5 0 7 6 .0 0 6 .2 0 1 .7 5 . 2 0 2 .6 1 7 .2 5 1 .2 2 1 2 6 .9 2 3 .6 7 10 .6 2 7 6 .2 5 3 .7 8 1 1 .3 3 2 0 1 .1 9 7 .1 0 2 1 .2 9 . 3 3 .1 0 2006 13 8 .6 6 ‘ 6 .9 2 5 .6 6 7 6 .0 7 6 .6 7 8 .2 2 21 6 .7 1 8 .3 5 1 6 .0 2 2016 167 .6 1 6 1 0 .6 5 6 1 5 .6 0 6 7 3 .0 0 6 7 .0 6 6 2 .2 6 6 2 1 7 .6 9 6 12 .6 1 6 1 7 .6 1 6 V ' 2017 13 6 .1 6 7.61 6 .1 3 76.0fl 6 .9 6 6 . 2 3 21 0 .2 1 8 .9 2 1 0 .3 3 3039 1 2 6 .6 6 6 .6 0 7 .5 7 ’ 5 2 .8 9 6 .3 3 6 .9 6 107 .33 7 .0 9 1 2 .5 5 • . 5001 1 2 7 .3 9 7 .1 5 6 .6 1 6 3 .2 8 6 .6 6 6 . 5 7 1 9 0 .6 7 8 .5 3 9.21 ; 5026 1 2 7 .2 6 9 .7 6 6 .7 6 . 66 .73* 6 .6 6 3 .1 2 19 1 .9 9 1 1 .7 8 7 .8 9 6022 12 7 .0 6 ■ 1 1 .0 0 6 .9 6 ‘ 6 0 .6 8 7 .3 6 6 . 9 7 1 8 7 .9 2 1 3 .2 3 1 1 .9 6 1 3 2 .0 0 2 .9 1 8 .7 3 6 7 .8 5 2 .3 5 i 7 . 0 6 1 9 9 .8 8 5 . 0 0 15.01 J i \ 1 3 0 .6 0 6 2 .5 3 6 7 .5 8 6 6 7 .6 5 6 2.666. 1 7 .9 9 6 1 1 9 8 .2 9 6 5 .1 6 6 1 5 .6 9 6 N I A 129 I UNIVERSITY OF SOUTHERN CALIFORNIA NUCLEAR PHYSIC'S LABORATORY • STATISTICAL EVALUATION OF DATA ACCUMULATED FOR THE F 1 9 IP ,H E 3 I REACTION o 0 - e x c i t e d STATE I - EXCITED STATE 10 ♦ I ) EXCITED STATE 0 ■ . t h e ta RUN SIGMA ERROR DELTA SIGMA ERROR DELTA SIGMA ERROR DELTA COM M3. iHHft) INHBI AVG IMHB1 (HHB1 AVG 1 MMB I 1 MMB ) AVG 3 5 ,8 3 3017 1 2 5 .9 7 7 . 5 0 8 .7 2 • 5 0 .9 3 4 .4 1 4 .8 4 1 7 6 .9 0 8 .7 0 1 3 .5 3 o 3018 • 10 9 .0 2 7 .2 8 8 .2 3 4 1 .6 7 4 .2 2 4 .4 3 1 5 0 .6 9 8 .4 2 1 2 .6 7 •' ' ‘ 1 1 7 .2 5 8 .4 7 2 5 .4 0 4 6 .0 9 4 .6 3 1 3 .8 9 1 6 3 .3 7 1 3 .1 0 3 9 .2 9 3 8 ,5 6 ‘ 2023 9 7 .2 6 5 .9 6 3 .4 9 35 .1 1 3 .5 0 3 .2 5 1 3 2 .3 7 6 .9 1 0. 73 4013 9 7 .7 2 6 .9 6 3 .0 2 3 2 .3 7 3 .7 6 0 .5 2 13 0 .1 0 7 .9 2 3 .0 0 o ' . ' 5004 1 1 2 .3 6 * 7 . 6 0 * 1 1 .6 1 * , 2 8 .9 5 * . 3 .5 6 * 2 . 9 t * 1 6 1 .3 1 * 8 . 3 9 * 8 . 2 1 * 5023 9 8 .0 7 8 .1 3 2 .6 7 30.51 4 , 3 0 1 .3 4 1 2 8 .5 8 9 .2 0 6.51 • ‘ 1 0 0 .7 5 3 .4 9 1 0 .4 8 . 3 1 * 8 6 1.3 9 4 .1 6 1 3 3 .1 0 2 .6 8 8 .0 3 o ' 1 9 7 . 6 0 * 0 . 2 3 * 0 . 7 0 * 3 2 . 9 7 * 1 .3 3 * 3 . 9 8 * 1 3 0 .7 1 * 1 . 1 0 * 3 . 2 9 * ! ;■ o * 4 1 ,2 7 5005 6 6 .8 7 5 .7 9 2 .1 3 21 .6 1 3 .1 7 2 .7 8 8 8 .6 8 6 . 6 0 0 .2 6 5022 7 1 .2 9 6 .0 0 2 .2 9 1 6 .9 2 2 .7 6 2 .1 1 8 8 .2 1 6 .6 1 0 .2 6 i " *. :* : O • ‘ 6 9 .0 0 ‘ 2 .2 1 6 .6 3 19 .0 3 2 .4 2 7 .2 7 8 8 .6 5 0 .2 6 0 .7 1 4 3 .9 8 2007 ■ 4 5 .5 0 2 .8 9 5 .6 6 /! i 9 . 3 i 1 .8 0 2 .6 7 6 6 .8 1 3 .6 0 3 .5 0 : o . 3042 5 6 .3 9 3 .1 0 5 .2 3 • 16 .0 5 1 .4 6 0 . 6 0 7 2 .6 3 3 .6 3 6 .1 3 4014 5 3 .5 4 4 .4 4 2 .3 7 1 4 .8 6 2 .2 4 1 .7 8 6 8 .6 0 6 .9 7 0 .0 9 5002 5 1 .4 6 4 .1 4 • 0 .2 9 15 .7 2 2 .2 0 0 . 9 3 6 7 .1 7 6 . 6 9 1 .1 6 o; * ‘ t 5 1 .1 6 2 .6 4 7 .9 3 1 16 .6 4 0 .9 5 2 .8 6 6 8 .3 1 1 .8 2 5 .6 7 ! * \ . ■' 6 6 .8 9 3043 3 3 .4 7 2.41 3 .3 6 1 1 .4 6 1 .3 2 0 .5 5 6 6 .9 3 2 .7 5 3 .9 5 1 6041 4 8 .0 3 4 . 4 0 1 1 .2 0 1 3 .6 0 2 .2 4 I.S B 6 1 .6 3 6 .9 6 1 2 .7 5 0 . ' i ’ 3 6 .8 3 6 .1 3 1 8 .4 0 12.01 0 .9 3 2 . 8 0 6 8 .8 8 7 .0 9 2 1 .2 8 i • f . 6 9 ,3 8 , 2008 3 5 . 9 8 * 2 . 3 0 * 4 . 5 3 * 1 3 .7 1 * 1 .3 7 * 0 .8 3 » 6 9 . 6 9 * 2 . 6 8 * 5 . 3 1 * i o ■ ’ 3044 30 .8 6 2 .2 0 0 .5 9 1 1 .9 7 1 .3 3 0 .9 1 6 2 .8 3 2 .5 7 1 .5 5 v • * . ’ 4015 2 6 .9 5 * 2 , 4 6 * 4 . 5 0 * 1 2 .3 6 * . 1 .6 1 * 0 . 5 2 * 3 9 .3 1 * 2 . 9 5 * 5 . 0 7 * . . f 5003 3 5 .5 0 4 .2 4 4 .0 5 12 .7 6 2 .4 9 0 .1 ? 6 8 .2 6 6 .9 2 3 .8 9 0 . 5006 3 0 .5 8 . 2 .0 9 0 .8 7 1 2 .3 5 1 .2 5 0 .S 3 6 2 .9 2 2 .6 6 1.65 5020 3 1 .3 7 2 .2 5 0 .0 9 14 .1 6 1 .4 3 1 .2 9 6 5 .5 3 2 . 6 7 1 .1 5 . ■ 1 . 3 1 .4 5 1 .2 7 3 .8 0 1 2 .8 8 0 .3 7 1 .1 2 6 6 .3 8 1 .6 9 6 .6 8 o ; ■ . . ; •' * ' ' 3 1 .2 9 * . 0 . 7 4 * 2 . 2 3 * 1 2 ,7 4 * 0 . 5 0 * 1 . 5 1 * 6 6 . 0 7 * 0 . 9 9 * 2 . 9 7 * C . • * . 5 2 .0 7 5021 3 0 .5 4 -3.07 0 . 0 17 ,7 4 2 .2 9 0 . 0 ' 6 8 .2 8 3 .8 3 0 . 0 3 0 ,5 4 • 3 . 0 7 9 . 2 0 . 1 7 ,7 4 2 , 2 9 * 6 . 8 6 6 8 . 2 S 3 . 8 3 1 1 . 6 8 ■ o I. \ ■ O . UNIVERSITY OF SOUTHERN CALIFORNIA NUCLEAR PHYSICS LAB. ORATOR^ O - ■ 1 STATISTICAL EVALUATION OF OAT A ACCUMULATED FOR THE F19IP»HE31 REACTION 0 - EXCITED STATE I - EXCITEO STATE • 10 ♦ I I EXCITEO STATE THETA RUN s ig m a ERROR OELTA SIGMA ERROR DEL 7 A s ig m a ERROR DELTA C3N N3. 1 MMB I (MMB) AVG (MMB) IMMBI AVG (MMB) (MMB) AVG 5 4 .7 5 2009 20 .5 2 I . 01 1 .1 8 1 7 .3 2 1 .3 6 0.781 4 5 .8 4 2 .2 6 1 .9 9 2011 . 3 0 .8 4 2 .0 0 1 .1 3 1 9 .8 7 1 .5 3 1 .7 6 5 0 .7 1 2 .5 1 2 .8 7 4016 2 9 .6 0 2 .8 1 O . l l 18 .5 9 2 .1 0 0 .4 8 4 8 . 19 3 .5 6 0 .3 5 5007 3 0 .1 3 2 .1 6 0 .4 3 17 .0 5 1 .5 6 1 .0 6 4 7 .1 8 2 .6 6 0 .6 6 • 2 9 .7 0 0 .5 4 1 .6 3 18.11 0 .6 8 2 .0 3 4 7 .0 4 1 .1 2 3 .3 5 ■-* ■ 3 7 .4 2 1 . 3050 • 2 7 .3 4 2 .4 1 0 . 0 1 7 .8 9 1.91 0 . 0 ! 4 5 .2 3 3 .0 7 0 . 0 • 2 7 .3 4 2 .4 1 7 .2 2 17 .8 9 1.91 5.72, 4 5 .2 3 3 . 0 7 9 .2 1 6 0 . 0 0 ’ • 2013 3 0 .7 3 2 .2 0 0 .9 5 10.08 1.7 4 i 1 .0 3 4 8 .8 1 2 .8 7 1 .9 6 4017 2 8 .5 7 2 .0 5 1 .2 2 16.61 1 .5 0 0 .4 4 4 5 .1 8 2 .5 5 1 .6 7 5008 • 3 0 .3 4 2 .2 9 0 .5 6 . 1 6 .6 6 1 .6 3 0 .3 9 4 7 .0 0 2.8 1 0 .1 5 2 9 .7 8 0 .6 8 • 2 .0 5 , 1 7 .0 5 0 .4 6 1 .3 9 4 6 .8 5 1 .0 6 3 .1 7 ■V. 6 3 .3 7 2026 2 9 .2 9 2 .6 5 1.56 ‘ 9 .6 8 1 .4 7 0.45' 3 8 .9 7 3 .0 3 1 .0 6 4019 2 5 .9 3 1 .9 4 1 .7 9 ' .10.4 8 1 .1 7 0 .3 5 3 6 .4 1 2 .2 6 1 .4 9 n • 5009 ’ 2 9 .1 9 2 .4 0 1.4 6 1 0 .0 5 . 1 .3 6 0 .0 8 3 9 .2 4 2 .7 6 1 .3 3 2 7 .7 2 I . 16 3 .4 9 •' 1 0 .1 3 0 .2 3 0 . 7 0 3 7 .9 0 0 .9 5 2 .0 6 r~ 7 0 .6 2 2027 . 2 3 . 1 6 * - 2 . 0 2 * 2 . 7 5 * 4 . 7 5 * 1 .2 8 * 0 . 5 1 * 2 7 .9 1 * 2 . 3 9 * 3 . 1 4 * 4020 2 0 .4 4 1 .7 3 0 .0 3 3 .9 3 0 .7 2 0 .3 1 2 4 .3 7 1 .0 0 0 .4 0 ! 5010 20 .9 2 2 .0 1 ' 0 .5 1 3 .5 7 0 .8 2 0 . 6 7 2 4 .4 8 . 2 .1 7 0 .2 9 5011 t 7 . 79 1 .0 0 2 .6 2 5 .4 3 0 .9 7 1 .1 9 2 3 .2 2 2 .0 4 1.55 • ' 2 0 .4 1 1 .0 9 3*25 4 .2 4 / 0 .4 1 1 .2 2 2 4 .7 7 0 .9 3 2 .7 9 o 1 9 .6 6 * 0 . 9 7 * 2 . 9 1 * 4 . 1 7 * 0 . 5 1 * 1 .5 4 * 2 4 .0 3 * 0 . 4 0 * 1 . 2 0 * • 7 5 .8 3 2029 1 2 .4 5 1 .1 3 0 .3 8 2 .1 3 0 .4 4 0 . 4 4 1 4 .5 8 1 .2 2 0 .8 9 i, 4021 1 2 .7 9 1 .0 4 0 .0 4 3 .2 5 0 .5 0 0 . 6 8 1 6 .0 4 1 .1 5 0 .5 7 5012 1 3 .3 0 1 .1 9 0 .4 7 2 .4 5 0 .4 9 0 .1 2 1 5 .7 5 1 .2 9 0 .2 8 1 2 .0 3 0 .2 4 0 .7 1 2 .5 7 0 .3 3 0 .9 9 1 5 .4 7 0 .4 5 1 .3 5 : c 0 0 .9 9 2030 9 .7 6 0 .9 4 0 .2 3 3 .0 6 0 . 6 3 0 . 0 . 12 .0 3 1 .1 3 0 . 0 5014 9 .1 8 1 .1 8 0 .3 6 0 .0 0 . 0 0 . 0 i 0 . 0 0 . 0 1 2 .8 3 - ' , • 9 .5 4 0 .2 9 0 .8 6 3 .0 6 0 .6 3 1 .8 4 12*03 2 0 .5 1 6 1 .5 3 8 6 .1 1 . 2031 1 0 .1 7 1 .0 2 0 .6 0 4 .3 3 0 .6 5 0 .0 5 1 4 .5 0 1.21 0 .5 3 c 5015 9 .2 1 1 .5 2 0 .3 5 4 .1 2 I . 01 0 .2 6 1 3 .3 3 1 .0 3 0 . 63 5016 9 .0 8 1 .0 7 0 .4 9 4 .5 9 0 .7 5 0 .2 1 1 3 .6 7 1 .3 0 0 . 3 0 9 . 5 6 0 .3 7 1 . 10 4 .3 8 . 0 . 1 2 0 .3 7 1 3 .9 7 0 .3 4 1 ,0 2 o C U N I V E * S I r V O F S O U T H E R N CA L I F - OA N I A N U C L E A R PHY SIC S L A B O R A T O R Y o . STATISTICAL EVALUATION Of OATA ACCUMULATED * FOR THE F 191P«HE3I REACTION n 0 - EXCITEO STATE 1 “ EXCITEO STATE 10 ♦ 11 EXCITEO STATE t h e t a . RUN SIGMA ERROR OELTA SIGMA ERROR OELTA s ig m a ERROR OELTA C35 N3. IMMBI I MMB) AVG 1HMB1 (MM0I AVG 1 MMB 1 ( MMB 1 AVG 9 i ; i 9 2036 - 1 1 .1 6 1 .3 6 1 .6 6 6 .2 3 0 . 7 7 0 . 1 0 1 5 .6 0 1 .5 6 1 .5 8 5017 , 13 .9 0 1 .2 9 1 .3 6 - 6 .6 2 0 .7 0 0 . 0 8 * 1 8 .6 0 1 .6 6 1.6 2 12 .6 3 1.61 6 .2 3 * 6 .3 3 0 .0 9 0 . 2 7 1 6 .9 8 1 .5 0 6 . 5 0 9 6 .2 1 2032 1 1 .6 6 0 .9 2 . 0 .1 6 ■ 3 .3 6 0 .6 5 0 .0 5 1 6 .8 2 1 .0 2 0 .2 0 * 5010 . 1 2 .0 0 1 .6 6 0 .3 7 3 .5 5 0 . 6 0 0 . 1 6 . 1 5 .5 6 1 .6 8 0 .5 3 1 1 .6 2 0 .2 3 0 .6 9 3 .3 9 0 .0 9 . 0 . 2 7 1 5 .0 2 0 .3 2 0 .9 6 o o o o o l o t . I I , 20 1 3 , 5 .6 9 S. AT O.SS 0 . 0 0 . 5 6 1 .6 9 . 0 .7 1 O .IS 0 . 0 6 . 2 0 0 . 5 9 0 . 0 0 .7 1 0 . 1 0 ' 0 . 5 6 . 6 . 2 0 0 . 5 9 1 .7 8 — V' • s. i ! o ' ,.C • o ••• n i . 132. UNIVERSITY N U C L E A R OF S 0 U T H'E RN CALIFORNIA PHYSICS CABORATORV STATISTICAL EVALUATION OF QATA ACCUMULATED FOR THE’ F19IP»HE3I REACTION ; ■ O o ■ o .o ! .•n O J 2 - EXCITEO STATE 3 - EXCITEO STATE 10 ♦ 11 EXCITEO STATE " J ' 7 V THETA RUN" SIGMA ' ERROR OELTA " SIGMA' " ERROR O a T A SIGHA ERROR OELTA ' COM NO. t HMBI (MHR) AVG ( HMBI, (HMBI AVG 1HMB1 (HMBI AVG i]:............ . . n . t s . 303^ ^ 4 5 .2 8 9 .1 6 0 . 0 1 9 .8 7 * 6 .0 2 0 . 0 ... :087.74 4 5 .6 2 8 .3 7 4 5 .2 8 9 . 16 2 7 .4 7 1 9 .8 7 6 .0 2 1 8 .0 7 ' 8 7 9 .3 7 4 5 .6 2 13 6 .8 5 — . n q Q - * 7 '3 0 3 3 ’ 3 7 . 9 4 * ' 1 0 . 2 0 * 0 . 0 * ' 1 6 .2 2 * 6 . 6 4 * 0 . 0 • 6 6 1 .5 4 * : 4 6 . 1 6 * 50. 70* 3 7 .9 4 10 .2 0 3 0 .6 1 1 6 .2 2 6 .6 4 19 .9 2 6 1 0 .8 4 4 6 . 16 13 6 .4 7 ---------- ------------------- - 0 . 0 ♦ 0 . 0 * 0 . 0 • 0 . 0 • 0 . 0 • 0 . 0 * . ,6 1 0 .6 4 * 4 6 . 1 6 * 1 3 8 .4 7 * 1 6 .6 8 2021 2 9 .4 0 3 .9 9 6 .0 7 2 9 .8 4 4 .0 2 6 .3 6 4 7 9 .1 0 2b . 26 15. 50 . . . . . T 3023 1 3 .2 5 * 7 . 6 6 * 1 0 . 0 8 * 1 3 .2 1 * 7 . 6 4 * 1 0 . 2 4 * 5 3 0 .0 2 * 5 0 . 5 9 * 6 6 . 4 2 * • 3024 2 1 .6 9 5 .1 5 1 .6 3 2 0 .5 0 4 .8 7 2 .9 6 4 4 6 .7 4 2 5 .4 8 1 6 .8 7 * ‘ 3026 1 6 . 3 1 * 9 . 4 3 * 7 . 0 1 * 1 6 .2 7 * 9 . 4 1 * 7 . 1 8 * 5 9 2 .2 7 * 5 9 . 19* 1 2 8 .6 7 * ... ...... . 4004 2 6 .6 1 5 .6 8 . 5 .2 9 2 3 .0 5 5 .0 8 0 .4 0 4 4 8 .4 4 2 5 .2 1 15.17 4010 1 4 .7 2 4 .9 3 6 .6 0 2 2 .8 4 6 .1 4 0 .6 1 4 5 5 .4 0 2 9 ,9 2 8. 20 $029 2 6 . 3 9 . 6 .0 3 3 .0 6 19. 74 5 . 14 3 .7 1 4 4 9 .7 8 2 7 .4 9 13 .8 3 4 6023 2 4 .5 3 5 .6 8 1.21 2 9 .6 2 *" * 6 .2 4 6 .1 7 4 7 7 .9 4 28 .0 3 1 4 .3 4 2 3 .3 2 2 .2 2 6 .6 5 2 3 .4 5 1 .9 3 5 .7 8 46 3 .6 1 9 .0 7 2 7 .2 2 L------— ----------- -i , . 2 4 . 4 1 * 2 . 3 5 * . 7 . 0 4 * 2 4 . 5 5 * ..........1 .8 8 * 5 . 6 5 *____ 4 6 1 .3 1 * . 6 . 4 5 * 1 9 .3 5 * 1 9 .6 5 • 2022 1 5 .2 3 2 .4 8 0 .3 1 2 6 .6 8 3 .3 4 5 .6 6 3 7 6 .4 5 1 5 .9 7 2 2 .4 5 ' . 1 " i 3022 1 3 .9 2 2 .5 0 1 .6 2 2 3 .2 9 3 .4 3 2 . 0 8 3 4 4 .1 6 15 .1 0 9 .8 4 r • J 3029 1 2 .3 7 4 .0 3 3 .1 7 1 1 .1 0 3 .9 2 1 0 .1 1 34 1 .7 5 2 2 .3 1 12 .2 4 V , 1 • 1 6— ~ r~ j 3030 2 0 .2 1 4 .5 6 4 .6 8 2 4 .2 0 4 .9 9 2 .9 8 3 4 0 .4 2 2 0 .4 6 13 .5 7 « 5028 2 3 .0 2 5 .1 4 7 .4 8 1 9 .6 6 4 .7 5 1 .5 3 3 6 1 .5 0 2 2 .1 9 7 .5 0 ; 6020 1 5 .2 1 * 4 . 0 9 * 0 . 3 3 * 1 9 .5 1 * 4 . 6 4 * 1 .7 0 * 3 9 6 .6 1 * 2 2 .9 5 * 4 2 . 6 1 * 1 5 .5 4 1 .2 7 3 .8 1 2 1 .2 1 2 .3 8 . 7 . 1 4 ...... 3 5 3 .9 9 8 .0 3 2 4 .1 0 — r ■ > 1 5 . 5 8 * ' 1 . 5 1 * 4 . 5 3 * 2 1 . 4 6 * 2 . 8 2 * 0 . 4 7 * 3 4 9 .7 1 * 6 . 86* 2 0 . 5 9 * ; 'Z Z 'iZ Z . 1 . 2000 8 . 5 6 * ” 2 . 2 3 * 0 . 6 1 * 2 3 . 9 2 * 3 . 7 6 * 5 .8 9 * 3 0 0 .4 3 * 1 4 . 0 5 * 2 7 . 5 5 * ‘ : 3000 6 .3 1 3 .6 5 2 .8 6 1 0 .4 9 - 4 .7 0 7 .5 5 26 7 .1 1 2 4 .5 5 5 .7 6 3007 9 .0 8 1 .9 5 0 .0 9 1 8 .5 2 2 . 0 2 0 . 4 9 2 5 5 .3 9 1 2 .0 4 . 1 7 . 4 9 ■ 3009 5 . 5 9 * 2 . 1 2 * 3 . 5 0 * 2 2 . 3 0 * 4 . 2 7 * 4 . 2 7 * " 2 4 3 .8 6 * 1 5 . 1 1 * 2 9 . 0 1 * • ; 3010 3 0 .0 6 4 .7 3 2 0 .6 9 H. 57 2 .4 9 9 .4 7 2 6 0 .3 9 1 5 .3 0 4 .4 9 3031 _ 8 . 4 2 * 2 . 19* 0 . 7 5 * 2 4 . 0 9 * 3 . 7 4 * 6 . 0 5 * 2 9 6 .6 4 * 1 4 . 6 0 * 2 3 . 7 6 * , 4001 8 .4 9 2 .7 0 0 .6 0 2 7 .9 6 4 .9 4 9 .9 3 2 7 5 .7 2 1 6 .5 7 2 .8 4 V v 4012 1 6 . 5 0 * 3 . 5 5 * 7 . 3 3 * 1 7 .2 1 * 3 . 6 3 * 0 . 8 3 * 2 4 2 .6 7 * 1 4 .5 0 * 3 0 . 2 1 * 5027 8 .5 1 2 .5 2 0 .6 7 2 1 .2 1 0 .6 9 0 . 0 0 2 8 3 .6 7 1 5 .0 2 1 0 .7 9 :. 1 ‘ . 6031 9 .1 5 2.9 1 0 .0 2 2 6 .4 7 - 4 .9 6 8 . 4 4 * ~ 2 8 3 .1 3 1 7 . 35 1 0 .2 5 j. *9.17 1 .4 3 4 .2 8 . 2 0 .4 2 1 .1 9 3 .5 6 2 7 2 .8 8 6 .2 5 1 6 .7 4 9 . 7 9 * 2 . 2 4 * 6 . 7 3 * 2 0 . 2 7 * 1 .6 1 * 4 . 8 2 * 2 7 4 .0 0 * 4 . 6 2 * 1 3 . 6 7 * 5 . • ' " UNIVERSITY NU C LEAR OF SOUTHERN C -A L I F 0 R N PHYSICS LABORATORY . STATISTICAL EVALUATION OF DATA ACCUMULATED FOR THE F 1 9 IP .H E 3 ) REACTION ZA.98 2 7 . TA AO 11 3 0 .5 0 3 0 1A 2 - EXCITEO STATE 3 - EXCITEO STATE SIGMA ERROR OELTA S IGMA ERROR OELTA 1 MMB) 1 HMBI AVG f MMB) IHM8) AVG 3 .1 0 1 .5 5 1 .3 5 1 5 .4 6 _ 3 .4 9 0 .9 1 5 .3 6 1 .9 0 0 .9 1 1 6 .0 3 3 .3 1 1 .4 8 4.31 1.91 0 .1 4 1 2 .1 9 3 .0 7 2 .3 6 6 .0 7 2 .1 5 1 .6 2 15 .1 3 3 .7 4 0 .5 6 4 .4 5 0 *6 6 1.9 9 1 4 .5 5 0 .9 2 2 .7 5 6 .2 5 1 .2 2 1 .6 9 15. 7! 1 .9 6 2 .6 3 4 .3 7 1 .4 4 1.4 2 1 3 .3 8 1.7 3 0 .5 1 1 2 .3 0 4 3 .9 3 4 4 .4 4 4 7 .4 1 4 3 .0 4 4 5 .4 6 4 9 .7 7 3 . 2 9 1 .8 2 1 0 .4 4 2 .6 0 2 . 4 3 7 .9 5 1 .0 7 3 .2 0 1 2 .8 8 1 .5 7 4 .7 2 7 .7 2 4 1 .1 2 4 3 . 3 7 4 _ 1 3 .7 0 4 . _ 1 .3 0 4 3 .9 0 4 I A 10 ♦ I I EXCITEO STATE b * _ ° i t o SIGMA i MMB I 2 6 7 .0 0 2 3 9 .5 9 2 6 8 .0 3 2 7 2 .1 2 2 6 0 .3 7 2 3 3 .7 4 212.86 2 64 .5 4 4 2 1 0 .2 3 2 1 8 .6 7 2 1 5 .4 5 * ERROR I MMB) 1 5 .5 5 1 3 .8 5 15 .3 2 1 5 .6 4 •7.83 9 .1 3 8 .3 3 1 9 .0 7 4 1 3 .1 0 8.66 7 .3 9 4 OELTA AVG 6 .7 0 2 0 . 79 7. 66 1 1 .7 5 2 3 .4 8 1 5 .0 7 5 .6 1 4 5 .8 7 4 8 .4 4 2 5 . 97 2 2 .1 6 4 n 3 3 .2 5 , D ! \T“ p;.. 0 t _ O'-' 1 i — b D 0 i “ P 3 5 .9 9 . 3 B . 7 2 . 200A 2016 2017 3039 5001 502A 3017 3018 12 .1 0 1 .9 7 2 .9 5 8 .7 6 1 .6 8 1 .0 9 ? 1 3 .1 0 9 .3 5 1 1 .9 1 7.61 1.81 1 . 54 7 .6 0 1.81 2 . 2 5 18 5 .7 7 9. 83 15. A2 8.7 3 1.21 0 . 4 2 . 1 2 .0 3 1 .4 3 2 .1 9 20 2 .4 1 7 .2 5 1.22 9 ,1 5 ____ i . I t . 3 .3 3 __ 9 .8 4 1*37 ♦ •11 2 0 1 .1 9 ___ 7 .1 0 2 1 .2 9 11 .7 8 1 .6 3 1 .3 4 6 .8 4 1 .2 3 1 .2 7 21 4 .7 1 8 .3 5 1A .82 1 3 .7 5 4 2 .5 6 4 3 .3 1 4 5 .9 5 4 1.6 6 4 0 .3 8 4 2 1 7 .4 9 4 1 2 . 6 1 * 1 7 .6 1 * 1 2 .7 8 1 .5 9 2 *3 4 4 .3 1 0 .9 0 1 .2 6 21 0 .2 1 9 .9 2 10 . 33 7 .8 2 1 .3 6 2 .6 2 _ 7 .5 7 1 .3 4 2 .0 0 187 .33 7 . 89 1 2 .5 5 1 0 .3 9 1 .7 4 0 .0 5 5 .8 8 1 .3 0 . 0 . 3 2 19 0 .6 7 8 .5 3 9.21 7 .9 7 2 .2 2 2 .4 7 3 .6 7 1.6 2 1 .9 0 19 1 .9 9 1 1 .7 8 7 .8 9 10 .4 4 0 .9 7 2 .9 2 5 .5 7 0 .6 0 1 .7 9 19 9 .8 8 5 .2 8 1 5 .8 5 1 0 .1 7 4 1 .0 2 4 3 .0 6 4 5 .5 3 4 0 .7 0 4 2 .0 9 4 198.294 5 . 5 8 * 1 6 . 7A* 7 .4 7 1 .5 7 1.0 9 4 .8 6 1 .2 6 2 .9 1 17 6 .9 0 8 .7 0 1 3 .5 3 1 0 .3 6 2 .0 2 1 .8 0 1 .1 5 0 .6 6 0 .8 0 15 0 .6 9 8.A2 1 2 .6 7 8 .5 6 1.4 0 4 .2 1 1.9 5 1.53 4 . 5 8 - 16 3 .3 7 1 3 .1 0 3 9 .2 9 ; 2023 . ___ 7 .7 3 l . A l . 0 .6 3 _ i . 2 .9 9 ___ 0 .8 7 0 . 0 2 1 3 2 .3 7 . . . 6 .9 1 0 .7 3 A013 8 . AS 1 .8 6 1 .3 5 . 3 .6 1 1.21 0 .6 1 130 .10 T. 92 3 .0 0 ' 500A 7 . 7 0 * 1 . 7 8 * 0 . 6 0 * A . 86* l . A l * 1 .8 5 * 1A 1.3 1* 8 . 3 9 * 8 . 2 1 * 5023 A . 58 1 .6 8 2 .5 2 1.71 0 .9 9 1 .2 9 128 .58 9 .2 0 A. 51 t " 7 .1 0 0 . BA 2 .5 2 3 .0 1 0 . 5 9 ' 1 .7 7 11 3 .1 0 2 .6 8 8 .0 3 i 6 .9 A * 1 . 1 3 * 3 . AO* . 2 . 7 0 * 0 . 5 2 * 1 . 5 7 * 1 3 0 .7 1 * 1 . 1 0 * 3 . 2 9 * 3 | : i * o; 134 b i s > D cv UNIVERSITY NUCLEAR OF SOUTHERN C A L I F 0 R H PHYSICS LABORATORY s t a t i s t i c a l e v a l u a t io n of oa ta acc um ulated FOR THE F 1 9 IP .H E 3 I REACTION I A 2 - EXCITEO STATE EXCITEO STATE THETA CON A I.A 5 A A .IT {” RUN*" s ig m a • ERROR OELTA SIGMA ERROR OEL TA NO. (MHO) IHHB) AVG IHHB) IHHB) AVG 5005 _ 5 . * 7 1 .2 7 0 .5 4 6 .0 4 1 .6 6 0 . 0 5022 4 .7 5 1 .4 4 0 .7 4 0 . 0 0 . 0 0 . 0 4 .0 1 0 .6 1 1.6 9 6 .0 6 1 .6 6 4 .9 6 200T 30 *2 A91A 5002 A 6 .8 9 ) 30*3 M A I A9.S 9 30AA AOI5 5003 5006 5020 5A .9 8 , 2009 2011 AO 16 ' 500T ST .66 > 3050 6 0 .3 2 , A017 5 .1 9 0 .8 4 1 .0 3 3 .5 9 0 .6 5 0 .5 8 3 .7 7 I . 10 0 .3 9 4 .4 7 1.1 3 0 .3 0 4 . 1 7 ____ 0 .3 9 1.1 8 5 . 7 0 0 .9 1 0 .5 0 4 .2 7 1 .2 4 0 .9 3 5 .2 0 0 .6 8 2 .0 5 4 .3 6 0 .7 3 0 .1 9 2 .9 8 4 0 .7 7 4 1 .2 0 4 5 .1 7 1 .5 7 1 .0 0 4 .1 0 0 . 6 9 0 .0 8 5 .1 7 0 . 84 1 .0 0 4 .1 8 0 .3 8 1. 14 4 .5 2 4 0 .2 6 4 0 .7 9 4 3.1 1 0 .4 9 0 .3 8 2 .1 4 0 .4 4 0 .5 9 3 . OS 0 .8 6 0 .3 5 3 .1 2 0 .6 3 0 .3 9 2 .7 3 _ 0 .2 7 0 . 82 2 .9 4 0 .7 4 - 0 . 0 2 .9 4 " 0 .7 4 2 .2 2 1 .9 2 0 .4 8 0 . 0 1*92 0 .4 8 1 .4 5 5 .5 7 0 .6 7 0 . 5 8 6 .2 3 . 0 .8 7 0 .0 8 7 .8 5 1 .5 9 1 .6 9 6 .1 3 1.3 2 0 .0 2 6 .1 5 0 *3 8 I . 15 A . 86 T.ll 5.3A 0.8A 1.6 0 0 .9 2 0 .A 8 1 .7 7 2 . 7 6 10 ♦ 11 EXCltEO STATE SIGMA ERROR OEL TA IHHB) IHHB) AVG 8 8 .6 8 6 .6 0 0 .2 4 88 .2 1 6 .6 1 0 .2 4 8 8 .4 5 0 . 2 4 0 . 71 64 .8 1 3 .4 0 3 .5 0 7 2 .4 3 3 .4 3 4 .1 3 6 8 .4 0 4 .9 7 0 .0 9 6 7 .1 7 4 .6 9 1.14 6 8 .3 1 1 .8 2 5 .4 7 AA.93 6 1 .6 3 A8.8A 2 .7 5 A.9A 7 .0 9 3 .9 5 1 2 .7 5 2 1 .2 8 5 .3 0 0 .8 1 0 . 6 3 4 2 .8 3 2 .5 7 1.5 5 5 .5 4 4 1 .0 6 4 0 .8 8 4 3 9 .3 1 4 2 .9 5 4 5 .0 7 4 4 .6 9 ~ 1 .4 9 0 .0 3 4 8 .2 6 4 .9 2 3 .8 9 4 .5 4 0 .7 3 0 . 1 2 4 2 .9 2 2 .4 4 1 .4 5 3 .8 7 0 .7 ? 0 . 7 9 4 5 .5 3 ? .6 7 1 .1 5 4 .6 6 0 .3 1 0 .9 3 4 4 .3 8 1 .2 2 3 .6 5 4 .5 3 4 0 .3 2 4 0 .9 5 4 4 4 .0 7 4 0 .9 9 4 2 .9 7 4 6 .0 6 3.7A 3 .7 9 0.0 A.5A 1 .8 3 1 .8 3 2 .1 6 2 .1 6 0 . 7 0 0 .5 9 0 .9 5 0.0 0 .6 3 0 .5 8 0 .5 8 0 .5 1 0 .5 1 1 .5 2 0 . 8 0 0 .7 5 0.0 1 .9 0 0.0 1 .7 5 0.0 1.5A 4 5 .8 4 2 .2 6 1 .5 9 5 0 .7 1 2 .5 1 2 .6 7 4 8 .1 9 3 .5 6 0 .3 5 4 7 .1 8 2 .6 6 0 .6 6 4 7 .8 4 1 .1 2 . . 3 .3 5 4 5 .2 3 3 .0 7 0 . 0 4 5 .2 3 3 .0 7 9 .2 1 4 5 .1 6 2 .5 5 " * 1 .6 7 4 6 .6 5 2 .5 5 7 .6 4 ' U L O ! uv 1 A ■ 1 ■ ■ 7 > . ; ■ ■ /“» • •' • * U N I V N E R S I T V U C I E U 0 F P H V S O U T H E R N CA S I C S L A B O R A L I F O T O R Y R 1 1 A o ! STATISTICAL EVALUATION OF DATA ACCUMULATED FOR THE F 1 9 IP .H E 3 I REACTION o * “ EXCITED STATE 3 - EXCITEO STATE 10 * 11 EXCITEO STATE 9 : • j O ^ t r T ~ v*;T THETA T ‘ COH 6 5 *6 3 r u n ' NO. 2026 6019 SIGH* ' I HMBI ______ _ 2.31 0 .6 8 0 .6 7 ERROR INN 81 0 .7 0 0 .2 6 0 .5 6 DELTA AVG 1 .6 3 0 .1 9 1 .6 9 SIGNA ERROR (NNBI (NHRI 1 .6 7 0 .5 6 3 .6 0 0 .6 7 2 .3 6 1 .0 5 OELTA AVG 0 . 8 8 1 .2 6 3 . 1 5 ' SIGMA INNS) 3 8 .9 7 3 6 .6 1 3 7 .9 0 ERROR 1 HMBI 3 .0 3 2 .2 6 1 .1 3 DELTA AVG 1 .0 6 1 .6 9 3 .3 9 —r - - - - - 1 • i ' O O 7 0 . 6 9 . 2027 6020 3011 1 . 0 5 * - „ 1 .3 9 2 .1 3 ... 1 .3 6 1 . 6 5 * 0 . 3 3 * 0 . 6 6 0 .6 0 0 .2 7 0 . 3 5 * 0 . 2 8 * 0 .0 5 0 .8 0 0 .8 2 1 . 0 6 * 3 . 0 5 * 0 . 5 7 * 2 .7 8 . 0 .6 1 1 .6 6 0 .5 2 2 .6 2 0 .6 5 2 . 1 2 * 0 . 5 6 * 0 . 6 3 * 0 . 3 6 0 . 7 8 1 .3 5 1 .6 9 * ... 2 7 .9 1 * 2 6 .3 7 2 3 .2 2 2 6 .7 7 . 2 6 . 0 3 * 2 . 3 9 * 1 .8 8 2 .0 6 1 .1 3 0 . 5 1 * 3 . 1 6 * 0 .6 0 1 .5 5 3 .6 0 1 . 5 2 * 1 .' c c . , . , _ _ . 7 6 . 1 1 ; 2029 6021 5012 0 .6 2 1 .8 7 0 .0 9 ; _____ 0 .2 6 0 .2 6 0 .3 7 0 . 0 9 0 .3 0 0 .3 7 1.6 2 0 .1 6 0 .8 9 . - 3 .5 6 .1 0 .5 7 2 .1 6 0 .6 0 0 .2 8 0 .1 6 - 0 . 7 6 ____0 . 6 9 2 .8 0 1 .6 2 C. 66 .. . 2 .0 6 .... _ 1 6 .5 8 1 6 .0 6 1 5 .7 5 1 5 .6 7 1.2 2 1 .1 5 1 .2 9 0 .6 5 0 .8 9 0 . ST 0 .2 8 1.35 r\- r< 8 1 . Z8 1 . . 2030 1 .0 2 ' 1.0 2 0 .2 6 _ 0 .2 6 0 . 0 0 .7 3 2 .7 2 0 .6 0 2 .7 2 0 .6 0 0 . 0 1.2 1 - - - 1 2 .8 3 1 2 .8 3 . 1 - IS 1 .1 3 0 . 0 3 . 3 9 C ■ ' 8 6 .6 0 1 5016 0 .5 7 0 .5 7 0 .2 8 0 .2 8 0 . 0 0 .8 5 1 .2 6 ^ 0 .6 0 1 .2 6 0 .6 0 0 . 0 1 .2 0 1 3 .6 7 1 3 .9 7 1 .3 0 1 .3 0 0 . 3 0 3 .9 0 o 9 1 .6 8 2036 5017 1.28 1 .5 8 0 .6 1 0 .6 1 0 .1 5 0 .1 5 0 .7 7 0 .3 2 0 . 0 0 . 0 0 . 0 0 . 0 1 5 .6 0 1 8 .6 0 1 .5 6 1 .6 6 1 .5 8 1.6 2 o 1 .6 3 0 .1 5 0 .6 6 0 . 7 7 0 .3 2 0 . 9 5 . 1 6 .9 8 1 .5 0 6 . 5 0 UNIVERSITY NUCLEAR OF SOU. THE' R N CALIFO PHYSIC, S LABORATORY o o o o. . o o o o n o. o o o . o f i o ■ n STATISTICAL EVALUATION* OF OATA ACCUMULATED FOR THE F I 9 IP .T 1 REACTION nit 0 - EXCITED STATE 1 - EXCITEO STATE 10 * 1) EXCITEO STATE THETA RUM sir, H A ERROR delta SIGMA ERROR OELTA i SIGMA ERROR delta COM M3* IMMft) (MMR) AVG IHMB1 IHHB) AVG • ( MMB 1 » MMB 1 AVG 13*90 6006 9 0 5 .8 2 * 6 6 .1 3 * 3 0 5 .5 9 * 2 8 6 .3 6 * 3 7 .6 5 * 1 2 2 .6 5 * 1 1 9 6 .1 7 * 7 6 .3 6 * 6 2 8 .3 6 * 6026 5 2 8 .2 5 61 .1 2 70 .9 7 • 162.28 2 1 .6 8 3 .6 3 6 9 0 .5 3 6 6 .6 9 7 5 .1 0 6039 5 6 7 .9 7 39 .6 5 5 1 .2 5 133.69 20 .1 6 3 2 .2 2 6 8 1 .6 6 6 6 .6 9 8 6 .3 7 5 9 9 .2 2 9 6 .7 2 2 9 0 .1 5 165.71 3 5 .3 0 1 0 5 .9 1 ; 7 6 5 .8 3 1 3 0 .9 5 3 9 2 .8 6 5 3 8 .6 7 * 9 .R S * 2 9 . 5 6 * 1 6 6 .8 6 * 1 6 .3 6 * 6 3 . 0 7 * , 6 8 5 .8 0 * 6 . 5 3 * 1 3 . 5 9 * 16*67 6003 5 3 6 .9 5 3 3 .8 8 32 .1 2 96*61 1 6 .0 6 1 7 .9 6 6 3 1 .5 6 3 6 .6 7 6 6 . 16 6009 53 2 .2 7 3 0 .1 6 2 7 .6 3 5 9 .8 9 to.o i 6 . 7 7 i 6 0 2 .1 6 3 1 .7 7 16. 7c» 6010 6 0 6 .1 5 * 3 6 .6 0 * 9 9 . 3 2 * 8 1 .7 6 * 1 2 .6 9 * 5 . 0 7 * 6 8 5 .6 9 * *8 .7 6 * 1 0 0 .6 9 * 5029 5 1 1 .5 2 3 0 .0 5 5 .6 8 6 9 .6 6 ■ 10.06 7 .0 2 ! 5 8 1 .1 6 31 . t 9 6, ’ s 6023 6 6 3 .2 3 * 2 9 . 0 7 * 6 1 . 6 1 * 5 7 .0 9 * 1 3 .5 9 * 9 . 5 8 * 5 1 0 .3 2 * 3 2 . 0 9 * 7 5 . 0 * * 6065 6 5 9 .5 5 2 6 .0 6 6 5 .2 9 9 0 .0 3 16.17 1 3 .3 6 5 6 9 .5 7 2 9 .6 5 35.8? 5 0 6 .8 3 2 2 .6 7 6 8 .0 0 7 6 .6 7 6 .6 6 1 3 .3 7 i 5 8 5 .6 0 2 3 .7 6 7 1 .2 9 5 0 6 .5 6 * 1 8 .7 5 * 5 5 . 2 6 * 7 7 .3 5 * 6 .2 0 * 1 8 .6 1 * ) 5 8 6 .9 3 * 1 6 .9 6 * 5 0 . 8 1 * 1 9 .6 6 5028 6 8 2 .1 7 2 8 .0 1 3 0 .5 6 1 0 6 .1 5 1 3 .6 3 1 3 . 1 0 | 5 8 8 .3 2 3 1 .0 6 2 5 .2 0 6020 6 2 1 .9 5 27 .6 1 2 9 .6 6 116.01 1 6 .6 6 6 .7 7 5 3 5 .9 6 3 2 .2 5 2 7 .1 6 6 5 1 .6 3 30.11 9 0 .3 3 10 9 .2 5 3 .8 6 1 1 .5 2 1 5 6 3 .1 2 2 6 .1 6 7 6 .6 9 22 .2 1 6002 6012 5027 6031 3 0 1 .3 5 3 8 6 .6 0 3 7 3 .7 9 3 6 1 .7 5 3 6 1 .6 0 2 5 .9 1 2 0 .6 2 1 9 .9 3 2 3 .1 1 1 7 .1 6 6 0 .2 5 2 5 .0 0 1 2 .1 9 0 .1 5 5 1 .6 1 19 6 .5 6 161 .90 169 .86 133.51 1 65.10 . 2 1 .5 6 12.06 1 1 .6 2 16.90 1 0 .0 3 6 1 . 6 7 j 6 .8 0 5 . 2 5 1 2 1 .5 9 3 0 .0 9 | 6 9 7 .9 1 5 6 8 .5 0 5 2 3 .6 6 6 9 5 .2 6 5 2 0 .8 0 3 3 .7 0 2 3 .8 8 2 3 .0 7 2 7 .5 0 12 .3 8 2 2 .8 0 27.71 2 .8 6 2 5 .5 6 3 7 .1 5 2 6 .9 7 5026 6021 2 9 2 .6 6 2 3 6 .8 7 262 .31 1 7 .5 8 1 6 .0 9 2 7 .7 9 3 0 .3 5 2 5 .6 6 8 3 .3 5 21 5 .1 7 25 0 .1 6 2 3 1 .2 7 16.01 1 6 .5 7 1 6 .8 9 15 .0 9 1 8 .8 9 : 5 0 .6 6 5 0 8 .0 3 6 8 7 .0 3 6 9 7 .9 8 2 2 .9 9 2 3 .1 0 1 0 .9 0 10. 85 1 0 .9 5 3 2 .7 0 2 7 .7 3 6011 502 5 6030 1 9 9 .5 6 2 3 9 .1 6 1 7 6 .6 7 . 2 0 5 .9 9 1 6 .8 6 1 6 .8 6 1 7 .7 0 1 8 .2 2 5 .6 3 3 3 .1 5 2 9 .5 2 5 6 .6 7 22 6 .0 2 26 7 .0 3 2 3 0 .8 3 2 3 6 .5 6 17 .9 7 1 7 .0 9 1 9 .1 9 7 .0 3 '1 0 .5 2 ! 1 2 .6 6 i 3 .7 2 ‘ 2 1 .0 8 , 6 2 3 .5 8 6 8 6 .1 5 6 0 7 .2 9 6 6 1 .0 6 2 6 .6 3 ? 3 .9 9 2f • 10 2 6 .2 5 1 7 .6 8 6 5 .0 9 3 3 .7 6 7 2 .7 6 3 3 .2 3 5026 ' 6022 / 1 90 .33, 167.61 1 6 1 .1 0 1 3 .8 6 1 6 .7 6 1 1 .1 5 9 .2 2 1 3 .6 9 3 3 .6 6 19 5 .6 7 2 2 7 .8 6 2 3 7 .0 6 1 6 .0 2 1 8 .7 6 1 5 .5 3 1 1 1 .6 1 ! 2 0 . 7 6 | 6 6 . 6 0 ! 3 8 5 .6 0 3 9 5 .6 8 3 6 9 .6 8 19.71 2 5 .1 6 6 .7 0 3 .6B 6 .0 0 1 6 .1 0 O UNIVERSITY OF SOUTHERN CALIFORNIA NUCLEAR PHYSICS LABORATORYi STATISTICAL EVALUATION OF DATA' ACCUMULATED fo r the F i v t P . n r e a c t io n 0 - EXCITEO STATE . 1 - EXCITED STATE 10 ♦ t l EXCITEO STATE r M THer* RUN SIGMA ERROR OELTA SIGMA ERROR DELTA SIGMA ERROR OEL TA com M3. IMMft) (HMBI AVG ( MHft) (HMBI AVG INMB1 IMM8I AVG 36 .7 1 4013 1 7 3 .9 7 1 0 .1 0 14 ,5 4 101 .75 7 .8 8 1 1 .9 6 , 2 7 5 .7 3 12.81 1 .4 6 0. 5004 1 5 1 .9 5 9 .3 4 7 .4 6 117.14 8 .0 3 3 .4 0 7 6 9 .0 9 12.31 5 .1 6 5023 1 5 3 .2 0 1 0 .4 8 6 .2 4 125 .99 9 .3 6 1 2 .2 6 2 7 9 .1 9 I * . 05 4 .9 4 r 15 9 .4 4 7 .1 1 2 1 ,3 3 1 1 3 .7 3 6 .9 9 2 0 .9 8 2 7 9 .2 5 2 .9 7 8 .9 0 4 1 .4 4 5005 14 4 .2 6 9 .5 8 4 .5 3 6 6 .2 4 6 .4 5 5 .5 3 2 1 0 .5 2 1 1 .5 5 9 .8 7 r 502? ‘ 1 3 5 .9 6 8 .7 6 3 .7 8 5 6 .8 7 5 .3 8 3 .8 4 1 9 2 .8 * 1 0 .2 8 7.8 1 1 3 9 .7 5 4 .1 4 12 .4 1 6 0 .7 1 4 .6 1 1 3 .6 3 2 0 0 .6 5 8.7B 2 6 .3 4 o / 4 4 .1 5 4014 10 6 .3 9 7 .0 9 9 .2 1 3 7 .7 2 4 .3 3 2 .1 5 1 * 6 .1 1 8.31 7 .8 3 5002 1 2 4 .9 2 7 .1 3 . 9 .3 2 3 4 .1 8 3 .4 9 1 .3 9 1 5 9 .1 0 7 .9 4 7 .1 5 r* 1 1 5 .6 0 9 .2 7 2 7 .8 0 3 5 .5 7 1 .7 3 5 .2 0 1 5 1 .9 5 7 .4 9 2 2 .4 6 o . 4 6 .6 7 6041 9 6 .5 4 6 .6 1 0 . 0 1 0 .6 7 2 .6 6 0 . 0 10 7 .2 2 7.31 0 . 0 9 6 .5 4 6.8 1 2 0 .4 3 1 0 .6 7 2 .6 6 7 .9 8 1 0 7 .2 2 7 .3 1 2 1 .9 3 c 4 9 .5 7 4015 8 3 .5 6 4 .9 2 4 .2 5 10 .9 0 1 .6 6 0 . * 3 9 * . * 6 5 .2 0 4.41 ' 5003 86.01 7 .0 0 8 .7 0 7 .9 5 2 .0 8 2 .5 2 9 5 .9 6 7 .3 0 5 .9 1 r 5006 77 .3 3 4 .0 5 1 .9 8 1 0 .4 2 1 .6 2 o.os 8 7 .7 5 4 .3 6 2 .3 0 . 5020 75.61 3 .9 9 3 .7 0 11 .6 1 1 .5 9 1 . 1 * 8 7 .2 2 4 . 3 0 2 .8 3 r ■ 79.31 2 . 4 7 7 .4 0 1 0 .4 7 0 .7 1 2 .1 2 9 0 . OS 2 . 0 7 6.2 1 5 2 .2 7 5021 6 8 .5 5 4 .9 0 0 . 0 L 9.56 2 .5 7 0 . 0 8 8 .1 1 5 .5 3 0 . 0 - 6 8 .5 5 4 .9 0 14-.69 19 .5 6 2 .5 7 T . T I 88.11 5 .5 3 1 6 .5 9 ■ - 5 4 .9 5 4016 6 4 .3 4 4 .8 3 4 .0 2 3 0 .3 6 3 .4 9 1 .1 3 9 * . 7 0 5 .9 5 5 .1 7 5007 5 6 .2 4 3 .4 6 2 .0 9 2 6 .7 2 2 .3 6 0 .5 2 ' 8 6 .9 6 4 . 2 0 2 .5 8 6 0 .3 3 2 .9 0 8 .6 9 2 9 .2 3 0 .7 6 2 .2 9 8 9 .S 3 3 .6 5 1 0 .9 5 o 6 0 .3 0 4017 4 1 .6 3 2 .5 8 1.6 1 4 4 .5 9 • 2 .6 9 0 . * 3 8 6 .2 2 3 .7 2 2 .0 6 5006 4 5 .2 6 2.9 1 ■ 2 .0 5 4 5 .5 3 2 .9 ? 0 .5 1 9 0 .8 1 4 .1 2 2 .5 3 4 3 .2 4 1.81 5 .4 4 4 5 .0 2 0 .4 7 l . * 0 8 8 .2 8 2 .2 6 6 .8 5 0 •6 5 .6 0 * 4019 4 3 .4 9 2 .7 8 2 .2 6 3 2 .1 7 2 .3 6 1 .0 9 7 5 .6 6 3 .6 4 I . 01 O 5009 3 8 .7 0 2 .9 1 2 .5 0 3 4 .7 2 2 .7 4 l . * 7 7 3 . * 3 4 . 0 0 1 .2 2 4 1 .2 0 2 .3 9 7 .1 ? 3 3 .2 5 1 .2 6 3 .7 9 7 * . 6 5 1.11 3 .3 3 UNIVERSITY OF SOJTHERN C U I F ] M II NUCLEAR PHYSICS LABORATORY STATISTICAL EVALUATION OF OATA ACCUMULATED FOR THE F 1 9 IP . T I RFACTI ON 0 - EXCITFO STATE . I - EXCITEO STATE 10 ♦ 11 EXCITCO STATE th e t a RUN SICMA ERROR OELTA SIGMA ERROR OELTA SICMA ERROR DEI 'A C31 N3. (MM8I IMM8I AVG (MM8I IMMBl AVG INNB) IMMBl AVG 7 0 .8 6 6020 36 .6 9 2 .5 5 0 .6 2 1 9 . 9R 1 .8 7 1 .7 0 5 6 .6 7 3 .1 7 0. 73 S010 6 0 . 90 3.01 3 .5 9 16.38 1 .9 0 1 .9 0 5 7 .2 8 3 .5 6 1 .3 6 SOU 33 .9 6 3 .3 6 3 .3 5 1 8 .6 0 2 .8 0 0 .3 2 5 2 .5 5 6 .3 6 3 . 39 37.31 1« 88 5 .5 3 18 .2 8 1 .1 5 3 .6 6 5 5 .9 6 1.31 3 . 9 , 7 6 . OB 6021 6 3 .6 5 2 .5 8 1 1 .7 7 6 .2 3 1 .0 6 0 .1 6 6 9 .8 5 2 .7 8 1 1 .7 9 SOI 2 7 6 .2 6 1 .7 8 5 .6 3 5 .9 5 0 .8 2 0 . 1 0 3 2 .1 9 1 .9 6 5 .6 7 31 • BA 8 .1 6 2 6 .6 3 6 .0 5 0 .1 2 0 .3 7 3 8 .0 7 8 .3 ? 2 6 .9 6 a i . 2 5 5016 2 0 .7 6 1 .8 3 0 . 0 2 .6 0 0 . 5 9 0 . 0 2 3 .1 3 1 .9 2 3 .0 20 .7 6 1 .8 3 5 .6 8 2 .6 0 0 .5 9 1 .7 8 2 3 .1 3 1 .9 2 5 .7 6 8 6 ,3 7 2051 12 .9 8 1 .2 3 1 .9 6 6 .2 6 0 .8 6 0 . 6 6 1 9 .2 3 1 .5 0 1 .5 6 S01S 1 5 .6 8 1 .9 9 0 .5 3 5 .7 1 1 .1 9 0 . 1 5 2 1 .2 0 2 .3 2 0.61 5016 17 .6 6 1 .5 2 2 .7 0 6 .9 6 0 .7 8 0 .6 2 2 2 .5 8 1.7 1 1 .6 0 16 .9 5 1 .6 6 6 .3 9 5 .5 7 0 .6 1 1 .2 6 2 0 .7 8 1 .0 7 3 .2 0 9 1 . AS 2056 11 .1 6 1 .7 6 2 .2 9 8.2 1 1 .6 0 0 .7 6 1 9 .3 7 2 .3 8 1.7 8 5017 16.76 1 .3 2 1 .2 9 7.21 0 . 9 0 0 . 2 6 2 1 .9 5 1 .6 0 0 .8 1 13.66 1 .7 7 5 .1 6 7 .6 5 0 .6 3 1 .2 6 2 1 .1 6 1 .2 0 3 .5 9 9 6 .6 7 2052 10.78 t . o t 1 .1 7 6 .9 8 0 .8 6 0 . 0 2 1 7 .7 6 1.3? 1 .3 0 5016 15.61 1 .7 5 3 .6 7 7 .0 6 1.2 2 0 . 0 6 2 2 .6 5 2 .1 3 3 .3 9 11 .9 6 2 .0 1 6 .0 3 7 .0 0 0 .0 2 0 .0 7 1 9 .0 6 2 .1 0 6 . 2 9 1 0 6 . SB 2055 6 . 6 8 0 .5 2 0 . 0 1.91 0 . 3 3 0 . 0 6 . 3 9 0 .6 2 0 . 0 6 .6 6 0 .5 2 1 .5 5 1.9 1 0 .3 3 . 1 . 0 0 6 . 3 9 0 .6 2 1 .8 5 O _U. N . I . V .6 R .S I. T .y NUCLEAR OF SOUTHERN .CU I FO.. PHYSICS LABORATORY STATISTICAL EVALUATION OF OATA ACCUMULATED FOR THE F 1 9 (P *T ) REACTION 2 - EXCITED STATE 3 - EXCITED STATE THETA COM 16*76 . RUN . NO* 6003 6009 6010 SIGH4 . . . ERROR OELTA IMMBI 6 5 *3 3 _ . 36*66 3 8 .8 9 6 38*67 38 *5 8 6 IMMBI 8 *5 3 6* 52 . 8*026 3 *0 6 AVG 6*66 6 *0 3 0 .2 2 6 9 .1 9 5 *1 6 6 . 1 5 . 6 7 * SIGMA I MMBI 0.0 6 2 .8 8 . . 66*666 50*01 6 2 *8 8 6 ERROR <HHB) 0.0 . 7 .2 6 * 10*606 7 . 22 7 . 2 6 * DELTA AVG * 0*0 7 .1 3 16*636 21*66 2 1*79 6 1 9 .5 5 ____ 5028 __ 3 9 .3 3 3 9 .3 3 6 .7 6 _ 6 *7 6 0 . 0 . 2 0 .2 3 . 2 7 .5 6 2 7 .5 6 5 .6 7 5 .6 7 0 . 0 1 6 .6 1 2 2 .3 3 502.7'", 11.01 11*01 3 .3 9 3 .3 9 o .o 1 0 .1 6 . 0 . 0 0 . 0 0 . 0 0 * 0 0.0 0 . 0 2 7 .8 8 ' 6011 .5025____ __ 2 2 .1 0 23*75 23*12 5 .2 5 6 *11 . 0 *8 0 - 1 .0 3 • 0 .6 3 _ . 2*61 0.0 2 1 .6 6 . 2 1 *6 6 0.0 6*11 6 . U 0.0 . 0.0 1 2 .3 6 '3 8 .9 0 ~ ~ “ ~ 6013 5006 . 5 0 2 3___11 . 1 3 .5 7 2 3 *1 7 .2 6 * 0 1 1 8*39 ™ 2 .3 6 3 .1 2 . . 3 * 9 0 ; 3*51 " 6 .8 1 ' 6 . 7 9 5 .6 2 :.r- . 10*56 21*11 9 .9 7 1 o . o __L__. 1 3 .6 8 2 .9 7 2 *0 2 0 * 0 3 .6 6 7.63" 3 *5 2 . .. 0.0 1 0 .9 8 "61 .66" 5005 5022 f 9* 82 1 0 .9 5 1 1 0 *3 6 ‘ 2 . 0 7 ' 2 .1 7 ... 0 .5 7 0 . 5 6 ” " 0 .5 9 1*70 . I 3 . 2 l “ :r? ~ 0.0 1 3 . 2 1 ____ 2 .6 1 0.0 2*61 0.0 0 * 0 7 .2 2 6 6 .3 8 ^___ 6016 J. 5002 ^ 9 .0 5 7 .6 5 8*13 — 1 .7 0 1 .6 6 0 .7 9 0 .9 2 _ 0 *6 8 2 .3 8 1 6 .6 6 15*69 1 5 .1 8 2 .1 8 2 .1 6 0 .5 3 0 .5 6 0 . 5 2 1 .5 6 6 9 .8 2 ....T ” 6015 '• 5003 5020 ‘ . 6 *3 3 6*6 5 2*61 3 .0 9 0 .9 3 1*68 0 .5 6 0 .6 8 1 .2 6 I * 56 . 0 . 6 8 . ,2.03 0.0 .... 0.0 12*98 _ __ 2 .6 9 9 .6 1 . . 1 .1 6 1 0 .2 0 • 0 .9 0 0.0 2 . 7 8 0 . 5 9 2 *7 1 10 + 1) EXCITED STATE SIGMA IMHB1 63 1 *5 6 6 02 *1 6 685*896 5 65 *6 0 5 66 *9 3 6 ERROR IMMBJ 3 6 .6 7 3 1 *7 7 3 8 .7 6 6 2 7 .9 3 2 1 *0 3 6 DEL TA AVG 6 6 *1 6 1 6 *7 6 1 00*696 6 3 *8 0 6 3 *0 8 6 3 1 *0 6 . 3 1 *0 6 588*32 5 6 3 .1 2 523 *6 6 520*80 6 2 3 .5 8 686 *1 5 6 61 *0 6 2 7 5 .7 3 26 9 *0 9 2 7 9 .1 9 2 7 6 *2 5 210*52 192*86 200*65 166*11 159*10 151*95 9 6 *6 6 9 5 .9 6 87 *2 2 2 3 .0 7 2 3 .0 7 2 6*63 2 3 *9 9 28 *8 6 2 5 *2 0 9 3 *1 9 2*86 69*21 1 7 .6 8 6 5 *0 9 8 6 ,5 2 12*81 1*68 12*31 5*16 1 6 .0 5 6 *9 6 2 .9 7 8*90 11*55 9 .8 7 1 0 *2 8 7*81 8 *7 8 . 2 6 *3 6 8*31 7*83 7 .9 6 7 .1 5 7 * 6 9 2 2 *6 6 5 *2 0 6*61 7 *3 0 5*91 6 *3 0 2 *8 3 I I . . . . . ._______ 1:___ .U D I V E R S I T Y OF S OU T H E R N C A L I F O R N I A N U C L E A R P H Y S I C S L A B O R A T O R Y , STATISTICAL EVALUATION OF OATA ACCUMULATED FOR THE F 1 9 IP .T I REACTION 2 • EXCITED STATE 3 - EXCITED STATE i o ♦ n EXCITED STATF THETA COM 6 0 * 5 8 - RUN S3. 5008 . SIGMA 1 HMBI '3 . 2 4 3*24 ERROR (MSB) 0 .6 8 0 .6 8 DELTA AVG . 0 . 0 . 2 .0 5 . . SIGMA IHM81 0 . 0 0 . 0 ERROR IMHBI 0 . 0 . o.o DELTA ! AVG 0 . 0 i 0 . 0 SIGHA IMMBI 9 0 .8 1 8 8 .2 8 ERROR IHMSI 4 .1 2 4 .1 2 DELTA AVG 2 .5 3 1 2 .3 6 6 5 .9 0 ____ 4019 ____5 .2 6 _ 5 .2 6 . . . 0 .9 2 0 .9 2 _ _ ..0 .0 . 2 .7 5 _____ 0 . 0 ■ 0 . 0 0 . 0 0 . 0 . 0 . 0 0 . 0 . ! 7 5 .6 6 7 4 .6 5 3 .6 4 3 .6 4 1 . 01 1 0 .9 3 7 1 .1 8 r " ‘ 4020 5011 4 .2 3 4 .3 3 0 .7 6 0 .8 5 ____ 0 .0 9 " 0 . 0 8 * 0 .1 0 . .. 0 .2 7 . 0 . 0 6 .4 9 ___u.. . 6 .4 9 0 . 0 1 .0 5 ____ 1 .0 5 0 . 0 0 . 0 . . 3 . 14 5 6 .6 7 5 2 .5 5 .. 5 5 .9 4 3 .1 7 4 .3 6 1 .6 9 0 .7 3 3 .3 9 5 .0 6 7 6 .6 0 ___ 1 “ 5012 2 . 87 2 .8 7 0 .5 3 0 .5 3 0 . 0 1 .5 8 5 .5 6 5 .5 6 0 . 7 6 0 .7 6 0 . 0 2 .2 9 _ 3 2 .1 9 3 8 .0 7 1 .9 6 1 .9 6 5 .8 7 5 .8 9 . i ___ 8 1 . SB ; 5014 “ 0 . 4 2 “ 0 .4 2 0 .2 6 0 .2 6 " 0 . 0 0 .7 9 5 .3 4 5 .3 4 i ” 0 .8 8 0 .8 8 0 . 0 2 .6 5 2 3 .1 3 2 3 .1 3 1.9 2 1 .9 2 0 . 0 5 .7 6 r- 8 6 .7 1 ; I 2031 5014 0 .6 6 ____ _ 1 .1 2 0 .8 0 0 .2 4 0 .3 7 0 .2 1 0 .1 4 0 .3 2 0 . 6 3 3 .2 4 . 2 .2 4 2 .7 3 : 0 .5 3 0 .5 2 . 0 .5 0 0 . 5 1 . _ 0 . 4 8 1 .4 9 | 1 9 .2 3 L 2 2 .5 8 2 0 . 7 8 . i 1 .5 0 1 .7 1 .1.50 ( ■ > 1 .5 6 1 .8 0 4 . 5 0

Asset Metadata

Creator Huber, Kent Alvin (author)

Core Title Investigation Of Two Nucleon Transfer Reactions In Fluorine-19 Using 45 Mev Protons

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program physics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,Physics, Nuclear

Language English

Advisor Cole, Robert K. (committee chair), Smit, Jan (committee member), Waddell, Charles N. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-574905

Unique identifier UC11362299

Identifier 7217474.pdf (filename),usctheses-c18-574905 (legacy record id)

Legacy Identifier 7217474.pdf

Dmrecord 574905

Document Type Dissertation

Rights Huber, Kent Alvin

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA