A Study of the O¹⁷(p, α)N¹⁴ Reaction Citation Brown, Ronald Edmund (1962) A Study of the O¹⁷(p, α)N¹⁴ Reaction. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ERAK-8Q61. https://resolver.caltech.edu/CaltechTHESIS:08012011-114449322 Abstract An experimental measurement of the absolute cross section for the O^(17)(p, α)N^(14) reaction has been carried out in the energy range from 490- to 1580-kev proton bombarding energy at a laboratory angle of 150°. Resonances were observed at bombarding energies of 518, 672, 747, 825, 927, 1096, 1101, 1247, 1274, and 1335 kev. Other level parameters were assigned where possible. The 747-kev resonance corresponds to a level in F^(18) at 6302-kev excitation which does not appear to have been previously reported. A calculation of the ratio O^(17)/O^(16) formed at equilibrium at various temperatures in the CNO-cycle in stars is made, and it is concluded that the terrestrial material which has been processed in the CNO-cycle underwent this processing at a temperature of about 17 x 10^6 °K. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Fowler, William A. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1962 Record Number: CaltechTHESIS:08012011-114449322 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08012011-114449322 DOI: 10.7907/ERAK-8Q61 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6560 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 19 Aug 2011 23:45 Last Modified: 27 Nov 2023 23:36 Thesis Files Preview PDF - Final Version See Usage Policy. 25MB Repository Staff Only: item control page

A STUDY OF TH,E 017(p,a)N 14 REACTION Thesis by Ronald E. Brown In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1962 ACKNOWLEDGMENTS The author wishes to express his gratitude and thanks to the entire staff and personnel of the Kellogg Radiation Laboratory for their kind assistance during the course of this work. He is es- pecially indebted to Professor William A. Fowler who suggested the problem and guided the work to its completion. Special thanks are also expressed to Barbara A . Zimmerman for her help with many of the computations. The author is also grateful for the graduate fellowships received from the National Science Foundation and for the financial support of this research by the joint program of the Office of Naval Research and the U. S. Atomic Energy Commission. ABSTRACT An experimental measurement of the absolute cross section for the 017(p.a)N 14 reaction has been carried out in the energy range from 490- to 1580-kev proton bombarding energy at a labo oratory angle of 150 . Resonances were observed at bombarding energies of 518. 672. 747. 825. 927. 1096. 1101. 1247. 1274. and 1335 kev. sible. Other level parameters were assigned where pos- The 747 -kev re sonance corre sponds to a level in F 18 at 6302-kev excitation which does not appear to have been previously reported. A calculation of the ratio 0 17 /0 16 formed at equilibrium at various temperatures in the CNO-cycle in stars is made. and it is concluded that the terrestrial material which has been processed in the CNO-cycle underwent this processing at a temperature of about 17 x 10 6 0 K. TABLE OF CONTENTS PART TITLE PAGE 1. INTRODUCTION. 1 II. APPARATUS 3 III. IV. v. . . A. General D iscussion. 3 B. Electrostatic Analyzer Energy Calibration 4 C. Magnetic Spectrometer Energy Calibration 5 TARGETS 7 A. Preparation. 7 B. Nickel Oxide Target Analysis 8 C. Advantages of Each Type of Target 12 EXPERIMENTAL PROCEDURE 14 A. Nickel Oxide Target Data. 14 B. Iron Oxide Target Data. . 19 ANALYSIS OF EXPERIMENTAL DATA 23 A. Conversion of Data to Cross Section vs Energy 23 1. The Energy Relation. . . . . . 23 2. Charge Exchange in the Target 26 3. Thick Target Formula. 27 a. General discussion. 27 b. Measurement of spectrometer solid angle 28 c. Application to nickel oxide target data 29 4. B. Thin Target Formula . 29 a. General discussion 29 b. Application to iron oxide target data 30 Error Analysis . . . . . . . . . . . . . . . 31 PART TITLE C. VI. PAGE Extraction of Nuclear Parameters . 34 1. Barrier Factors • 34 2. Angular Distribution Factors. 35 3. Broad Resonance Analysis. 36 4. Narrow Resonance Analysis 39 . D. Discussion . " . . E. Summary of Experimental Results. 44 47 THE ASTROPHYSICAL PROBLEM. 49 A. General Discussion . . 49 B. Cross Section Formula. 51 C. Choice of Reduced Widths 54 D. Calculat ion of Reaction Rate s 56 E. Discussion of Results 61 APPENDICES.. . . . 64 A. Calculation of Barrier Factors 64 B. Angular Distribution Factors 69 C. Energy Spread Introduced by Spectrometer Solid D. Angle . . . . . . . 72 Energy Straggling 74 REFERENCES 85 TABLES 88 FIGURES 100 -1- 1. INTRODUCTION In recent years theoretical work on the properties of the mass-18 system (Redlich, 1954, 1958; Elliott, 1955, 1958a, 1958b) has stimulated a good deal of work on the levels of F lying levels (Ajzenberg-Selove , 1959). 18 --especially the low- In this thesis the results of an investigation of several of the higher levels of F 18 by means of the . .IS reporte. d 0 17( p, a )N14 reachon our present knowledge of the F 18 Al so reporte d IS . an app1"lcahon 0 f level structure to an estimate of the . . star s. 0 17(p, a )N14 reachon rate In In the present experiment an excitation curve at a laboratory angle of 150 0 was taken with protons ranging in energy from 490 to 1580 kev. This covers the region of excitation in F 18 from 6 . 06 to 7 . 09 Mev . Below 1-Mev bombarding energy several very narrow, well isolated resonances were found, and above this energy several narrow anomalies superimposed on rather broad resonances were observed. vious work on the 0 17 (p, a )N 14 Pre- reaction had been carried out in the region of 1- to 3-Mev bombarding energy (Ahnlund, 1957). In the present work some structure near 1250 kev was seen that was not reported by Ahnlund (1957). The present work and the work of Ahn- lund (1957) are the only reported investigations of the 017(p , a)N reaction to this date . c entration (. 040/0) of 0 factors necessary. 14 This is presumably because the very low con17 in natural oxygen make s large enrichment Recently enrichments of 0 17 of up to 40/0 have become available . * * Weizmann Institute of Science, Rehovoth , Israel; Isomet Corporation, Palisades Park, New Jersey. -2Several other reactions have been used to investigate this region "F o f eXCl"t a t"Ion 10 18 • Th e F19(H e 3 ,a )F 18 " has b een use d to reactIon measure the level positions in F 18 (Hinds, 1959). Several of the angu- lar momentum and parity properties have been investigated by means of N 14 (a, a )N 14 elastic scattering experiments (Heydenburg, 1953; Kashy, 1958; Herring, 1958a, 1958b; Silverstein, 1960). Also inves- " 14 17 tIgated have been N (a,p)O (Heydenburg, 1953; Kashy, 1958; Herring, 1958a) and N 14 (a,y)F 18 (Phillips, 1958). In Section V-D some of these results will be compared with those obtained in the present experiment. The importance of the 0 17 (p, a)N 14 reaction in theories of ele- ment syntheSis in stars comes from its occurrence in the well-known carbon-nitrogen-oxygen cycle (CNO-cycle) in which it acts as a feedback into the main part of the cycle. The reactions occurring in the CNO-cycle are (Fowler, 1960) C 12(p, y)N 13 ('B + v)C 13 C 13(p, y)N 14 ~ N14(p,y)015(~+ v)N 15 N 15 (p,a)C 12 or (1/2200) N 15 (p, y)O 16 016(p,y)F17(~+ v)017 017(p, a )N 14 -2aKnowledge of the ratio of the a:mount of 0 16 to the a:mount of 0 17 formed in the eNO-cycle then depends on a knowledge of the cross sections for 016(p, y) which for:ms the 0 destroys it. 17 and for 0 17 (p, a) which This ratio is important in the light of a recent paper by Fowler, Greenstein, and Hoyle (1961) which discusses ele:ment formation in the early history of the solar syste:m. This thesis thus consists of two main parts. In Sections II through V we discuss the experi:mental determination of the . o 17 (p,a)N 14 cross sectlon and the assignment of level parameters to the various states in F18. Section VI deals with the astro- physical problem in the light of the suggestions of Fowler et al. (1961). -3- II. A. APPARATUS General Discussion The Kellogg Laboratory 2-Mv electrostatic generator was used to accelerate the protons for this experimental determination of the . . energy excltatlon curve f or 017( p, a )N passed through an 80 0 14 . Th e proton b eam was electrostatic analyzer into a target chamber. A signal taken from the horizontal slits at the bottom of the analyzer was used to regulate the generator voltage. The analyzer slits were set such that the energy resol u tion was about 0.20/0. A double focus- ing magnetic spectrometer with an equilibrium orbit radius of 10.5 inches (Snyder, 1950) was used to analyze the reaction products. The target surface defines the object plane; at the image plane a CsI crystal was mounted on a lucite light pipe, the optical contact being made with Dow Corning high vacuum grease. The other end of the lucite was sealed in the same manner to the surface of a Dumont 6291 photomultiplier. The output pulses from this photomul- tiplier were passed through a standard preamplifier and pulse amplifier . The output of the pulse amplifier was fed into a biased amplifier and then into a 10-cpannel pulse height analyzer. The 10-channel analyzer was set in the five-volts-per-channel mode and the biased amplifier allowed this 50-volt window to be positioned in the desired region of pulse heights. In order to integrate the beam current a capacitor of known capacitance was charged by means of a bank. of znercury batteries of known voltage. this capacitor. The beam current was then allowed to discharge At coznplete discharge the counting equipment was autoznatically gated off by a systezn of relays. In this way the total -4number of bombarding protons per run could be found . In order to avoid electron leakage to or from the target when the beam is on, two precautions were taken. The target was raised to a potential of 300 volts above ground in order to prevent electrons from leaving the target when the beam strikes 'it. Also, a screen at a potential of 300 volts below ground was placed at the entrance to the target chamber. This prevented electrons which were produced at the slit systems from finding their way to the target. This arrangement requires that 0.3 kev be subtracted from the proton energy E e at the electrostatic analyzer exit to obtain the proton energy E1B at the target surface . B. Electrostatic Analyzer Energy Calibration The voltage across the plates of the electrostatic ~alyzer is determined by tapping off a fraction of this voltage and reading it with a potentiometer; call this reading V . e Let M and Z be the mass and charge number of the particle with energy E through the analyzer. e which passes Following the derivation of Mozer (1956) but using the notation of Bardin (1961) we may write E H ere k e e =k e ZV e (1 + E e /2Mc 2 ) (1) is the calibration constant, and c is the speed of light. In order to determine k resonance in the F 19 e the gamma rays from the 873-kev (p, ay)016 reaction were observed by means of a sodium iodide crystal, photomultiplier arrangement. A thick CaF target was prepared by evaporation onto a Cu backing. The -5Cu backing consisted of Cu evaporated onto a clean glass microscope slide. The midpoint of the step in the thick target gamma ray yield was observed to occur at a setting of V :: 0.8686 decivolts. e assumed that this value corresponds to the resonant energy E for the F 19 (p, a'Y)O 16 . reactIon. It is res After a survey of the energy deter- minations of this resonance (Bondelid, 1959, and references therein) the following value was adopted: E res :: 872. 7 + O. 4 kev This is to be compared with the recent value of 872.5:: 0.4 kev adopted by Marion (1961) in his article on energy calibrations. rection for the target potential gives E solving Eq. (1) for k e Cor- = 873.0 + 0.4 kev, and e gives ke .. 1. 0047:: 0.0006 Mev/decivolt C. Magnetic Spectrometer Energy Calibration The magnetic field in the spectrometer is measured and regulated to one part in one thousand by means of a rotating coil fluxmeter and optical lever system (Milne, 1953). The current in the fluxmeter coil is determined by measuring the voltage across a precision resistor; call this voltage V m . If we again follow the derivation of Mozer (1956) and use the notation of Bardin (1961) we may write (2) -6where E m is the energy of a particle of charge number Z and mass M passing through the spectrometer, and k m is the calibration con- stant. In order to determine the spectrometer constant k m tion Cu(p, p)Cu was observed at a laboratory angle of 150 the reac0 • Thick Cu targets were prepared by evaporation onto clean glass microscope slides. Protons of approximately 1 Mev were scattered from the Cu and a target profile was taken (the term target profile will henceforth denote number of counts NO versus fluxmeter setting V m for a fixed bombarding energy E 1B ). The bombarding energy is known from the electrostatic analyzer calibration, and the energy of the protons which are elastically scattered from Cu atoms at the target surface can be calculated from kinematics. The fluxmeter setting corresponding to the midpoint on the target profile rise is taken to correspond to the scattered proton energy at the target surface. The small correction due to the target potential was again made. The result of several trials is k m = 384 100 + 200 Mev-mv 2 The determination of the solid angle of the spectrometer will be discussed in Section V -A. -7III. TARGETS A. Preparation The relative abundance of 0 17 in natural oxygen is about 0.040/0 so that a considerable enrichment of the 0 experiment. 17 is needed for the present Two types of targets were used. One was a 1/ 16-inch thick stainless steel disc which had been bombarded in a mass sepaI 14 17 rator WI°th 0 In th e f orm 0 f t h e Ion N 0 17 . 0 0 nonuniform target of 0 17 0 ThoIS resu It e d m a th In, O 0 This target was brought to the Institute from Sweden by Katarina Ahnlund for her investigation of the o 17 (p,a)N 14 reaction (Ahnlund, 1957). 0 Upon completion of her work here she kindly left; the target at the Institute. In what follows this target will be referred to as the iron oxide target. The other type of target used was a thick, uniform nickel oxide target. These targets were prepared by oxidizing clean, polished, is-mil thick nickel blanks in an induction heater. The oxygen gas used was obtained from the Weizmann Institute of Science, Rehovoth, Israel, and was of composition 3.97% 0 0 16 . 17 , 43.70% 0 18 , and 52.33% Their isotopic analysis has been assumed to be correct. At an early stage of the experiment a sample of oxygen gas enriched to 2.77% 0 17 and 71. 7% 0 18 was obtamed from the Isomet Corpora- tion, Palisades Park, New Jersey. 0 The alpha-particle yield at several bombarding energies was found to be in the ratio 3.97/2.77 for the two targets, thus lending support to the above quoted 0 c entrations . 17 con- Nickel was dec ided on as the element to be oxided with the enriched gas after a series of trials were made in oxiding different metals (using natural oxygen). The elements W, Cu, Ni , Mo, Ti, and stainless steel were all tried . Protons were then scattered -8from the oxided targets to determine the deg ree of oxidation and uniformity of the oxide layer as described in Section llI-B b e low. It was found that it was easiest to obtain a good NiO target, hence Ni was chosen. The oxide was found to be stable--no oxygen loss was detectable with the beam currents of the order of 1 "m icroamp or less and the beam spot size of the order of 1 mm x 2 mm used in this experiment. B. Nickel Oxide Target Analysis To obtain the absolute cross section one must know the number of reacting target atoms per cm 3 in the target. It is thus important to know what fraction of the nickel atoms in the target are oxidized. In order to detennine this a target profile was taken of i-Mev pro- tons elastically scattered from the target. From the profile both the degree of oxidation of the nickel and the thickness of the oxide layer can be determined. Such a profile along with a profile taken from a pure nickel target is shown in Fig. 1. The protons counted in both cases are those elastically scattered from the nickel atoms. The decrease in counts at the step in the oxided target is due to the influence of the stopping power of the oxygen. When the magnet setting is such that the observed scattered protons start coming from behind the oxided layer then the yield approaches that from pure nickel; thus the thickness of the oxide layer can be determined. Let the symbol [X] stand for the number of X atoms (or molecules) per unit volume in the target, and E X stand for the stopping c r oss section per atom (or molecule) for material X . quantitie s a and .... by We then define two -9- a - [NiO] - £ NiO (3) [Ni] = fL (£ Ni + £ 0) (4) It should be stressed that the symbol [Ni] is meant to include all the nickel in the target- -including that contained in the nickel oxide. Thus a measures the degree to which the oxide layer has been oxi- = 1 for pure NiO) and fL measures the error made in assuming that the additivity of stopping powers is valid (fL = 1 for no error). dized (a We now derive a formula which compares the yield at the step in the two cases. The same proton energy is used so that the Rutherford cross section is the same in both cases. For the same cross section, the scattered proton yield is directly proportional to [ Ni] t, the num ber of nickel atoms per cm 2 as seen by the spectrometer. For the case of elastic scattering and where the normal to the target surface bisects the angle between the incident particles and the outgoing particles we have [ Ni ] t oc: 1 where £ 1 is the stopping cross section per nickel atom for the incoming proton, £ 2 is the stopping cross section per nickel atom for the scattered proton, and E / E is the ratio of the scattered 1 Z energy to the incident energy as determined from kinematics (Brown, 1951). Let N be the number of counts at the step in the NiO target and N + D. N be the number of counts a t the step in the Ni target . Then we find -10[£ 1 (N + .6.N)(£ 2 / £ 1 + E/Ei] Ni = [£ 1 N (£ / £ 1 + E/E 1 )] NiO On asswning a = 1, fL (5) = 1, the terms £ 2/£ 1 + EzlE 1 are found to be very nearly equal in the two cases; thus this term will be canc e lled from the equation giving (N + .6.N)£ Ni =N£ T (6) where £ T is the stopping cross section per nickel atom in the oxided region of the target. £ T ::: [~:] It is given by ~::: [~i] {(I Ni) -[ NiO) )£ Nt [NiO) £ NiO} (7) On combining Eqs. (3), (4), (6), and (7) one finds the following relation between aand fL (8) From Fig. 1 we find N = 67 490, .6.N::: 21 900. We take the nickel and oxygen stopping cross sections from published curves (Whaling, 1958). Substitution into Eq. (8) then gives the following experimen- tally determined relation between a and fL. a::: 1 This relation is shown graphically in Fig. 2. cant region of the graph is for (9) 3.06 fL - 2.08 a ~1. The physically signifi- In order to determine target composition, a value for fL must be chosen. a, the It has been found e'lCperimenta1ly that for many compounds fL is close to unity, even at proton energies lower than those used in the present work (Reynolds, 1953). Also, Gray (1943) has pointed out t hat deviations from -11the additivity law should not amount to more than 10/0' Platzman (1952), however, believes that 1% may be an underestimate and that in some cases deviations up to even 5% may be present. must be pointed out, however, that £ £ °' ° It itself is not known; only the stopping cross section for the oxygen molecule, can be 2 measured. In the computation of £ NiO one assumes £ ° ='21 £ 02 and thus .... will reflect only the difference in binding in 02 and NiO. One might hope that this difference will be quite small; al- though Platzman points out that the valence bonding in molecules containing oxygen can vary rather strongly. Thus there is some uncertainty as to the exact value to use for..... An inspection of Fig. 2 shows that the data are consistent with the values a = 1, . . = 1. Also, investigation of several NiO targets of varying thickness showed that .6. N did not change. This suggests a =1, since if all the nickel were not being oxided one would expect a to be a function of the oxidation time (and hence of the thickness of the oxide layer). In the following analysis it will be assumed that a =1, .... =1. The stopping cross section for NiO will be computed from 1 £ NiO and £ Ni and i £ 02 =£ Ni + '2 £02 (10) will be taken from published data (Whaling, 1958). The thickness of the oxide layer can also be found from the scattering data. Figure 1 corresponds to an oxide layer thickness of 26 kev to I-Mev protons. -12C. Advantages of Each Type of Target At a given proton bombarding energy the peak alpha-particle yield from the iron oxide target was found to be about four times that from the nickel oxide targets. In all other respects , however, the iron oxide target was inferior to the nickel oxide. Since it had been used in a previous experiment the iron oxide target had carbon contamination on the surface. The number and distribution of 0 17 atoms in the target was a function of the target spot that was being bombarded. Since the iron oxide target was so thin (about 6 kev to i-Mev protons) it was necessary to take a complete target profile at each bombarding energy. This fact and the carbon build up during the bombardment made it extremely difficult to obtain any quantitative data with this target at the several very narrow resonances observed in this work. Relative cros s section measurements were made with the iron oxide target at several energies away from the very narrow resonances. particularly where the cross section was quite low and the low yield prohibited use of the less enriched nickel oxide targets. These relative cross sections were normalized to the absolute values obtained with the nickel oxide targets. Even though the yield was lower for the nickel oxide targets the majority of the data was taken with these targets. These targets are thick enough and uniform enough so that the thick target relations (Section V -A) can be used. Thus the magnet setting determines the number of target atoms per cm 2 as long as it is set to detect only particles produced completely in the uniformly oxided region of the target. In order to avoid undesirable effects due to carbon build up -13on the target its position relative to the beam can be shifted as often as desired. This is especially important when taking the data at the very narrow resonances. -14IV. EXPERIMENTAL PROCEDURE A. Nickel Oxide Target Data For this data the targets used were kept as clear of surface contaminants as possible ·. The ideal case was when the target was removed from the induction heater and placed immediately into the target chamber, which was then pumped down. not always followed , however. This procedure was Many runs were taken with targets which had been under vacuum for several days, either in the target chamber itself or in a separate vacuum jar. It was found that such a waiting period did not affect the experimental results. There are many protons produced by elastic scattering from the nickel in the target. These have an energy spectrum from zero up to an energy almost equal to that of the bombarding protons . The magnetic spectrometer, besides defining the reaction lamina in the target, also serves the very useful purpose of separating these scattered protons from the alpha particles under observation . The spectrometer was set to observe the doubly charged alpha particles from the 0 17 {p, ex )N 14 reaction. This reaction has a Q-value of 1.193 Mev as determined from recent mass tables (Everling, 1960). Protons of the same energy as that of the alpha parti- cles will pass through the magnet. At SLab = 150 the maximum 0 energy of the scattered protons becomes equal to that of the alpha particles at about 2-Mev proton bombarding energy. This critical energy becomes larger at smaller scattering angles. However, the Rutherford cross section increases rapidly as one moves to more forward angles. This would tend to greatly increase the num- ber of protons being counted due to scattering through the spectrometer. - 15. It was found that a scattering angle of 150 0 was a good compromise. Because of the low alpha-particle yield in this experiment both the entrance apertures and exit slits to the spectrometer were removed. Removing the entrance apertures allowed the full solid angle of the spectrometer to be used. Removing the exit slits in- creases the counting rate by making the energy resolution poorer- that is, the magnet is allowed to pas s a larger portion of the energy spectrum of the particles incident on it. In taking the data the target chamber entrance slits were first adjusted so that the proton beam hit the target surface in the center of the target chamber. 2-mm wide. The beam spot was about 1-mm high and The target was then positioned so that the normal to the target surface bisected the acute angle formed by the spectrometer (set to give a scattering angle of 150 0 ) and the incoming proton beam. All targets were always positioned in this manner for all the data taken in this experiment. The spectrometer was then set so that the lamina in the target from which the alpha particles were being counted was completely in the oxided layer and below the target surface to a depth of several kev to the bombarding protons. This setting was determined from Eq. (13) (to be discussed in Section V -Ai) and was checked by taking a target profile at several bombarding energies. This spectrometer setting is a function of the proton bombarding energy. For the beam current integration three polystyrene capacitors of nominal value 10 IJ.f each were connected in parallel. The voltage of the bank of mercury batteries used to charge this capacitor combination was measured with a precision voltage divider and potentiometer. This voltage was measured several tiIne s during a run -16as a check on the condition of the mercury batteries. Normally the voltage was close to 9.45 volts and could be measured to better than one part in five thousand. The total capacitance of the parallel com- bination was measured by first charging it by means of the mercury batteries--then discharging it with a 300-volt emf and some large resistors in series. The time necessary to accomplish complete discharge was then recorded. C may be calculated. From this data the total capacitance The RC time constant was long compared to the discharge time so that the correction to the capacitance for non~iform discharge current amounted to a of three trials was C = 29.3:: 0.1 fL£ . little over 1%. The result Each integration, then, amounted to a total charge deposited of about 276 microcoulombs, and took approximately ten minutes. Aiter every two and sometimes three integrations the target was shifted so that the proton beam would bombard a clean spot. Even after one integration carbon build up was clearly visible on the target surface." By shifting every two or three integrations it was found that the data taken at the narrow resonances were quite repeatable. This would indicate that the energy shift introduced by the carbon build up from three integrations on a fresh target spot is probably less than 0.5 kev. As discussed in Section II the alpha-particle pulses were fed into a 10-channel analyzer. It was not strictly necessary to use this analyzer at energies where the cross section was highest but it was extremely convenient and all the nickel oxide target data were taken using it. It was a simple matter to set the pulse amplifier gain and biased amplifier bias so that the peak of the pulse height -17distribution fell somewhere in channel five to seven . This could be done empirically at an energy where the yield was relatively high and then the known response of cesium iodide to alpha particles could be used to determine the settings at other energies (BashIn this way no very accurate bias settings were needed. kin , 1958). The work involved in recording the data was greater than had a single scaler been used; however, it was felt that the increased confidence in the data was well worth the effort. The solid histograIIl in Fig. 3 shows a sample pulse height spectrum for one integration obtained at a bombarding energy of L 342 Mev. To keep a check on the background a nickel oxide target was made using natural oxygen. At any time during a run the natural target could be moved into the beam and a background integration taken. The dashed histogram in Fig. 3 shows such a background integration. A spread of five to seven channels was always sufficient to encompass the alpha-particle spectrum. The remaining channels were simply discarded as far as the det~rmination of the number of alpha counts and of background counts was concerned. For instance in Fig. 3 channels one, two , three, and surplus were discarded. It was found that below 1. 50-Mev bombarding energy the background ranged from about five to twenty counts per integration depending on the bombarding energy and the general laboratory background at the time. This background is attributed both to laboratory back- ground and to counts due to protons which scatter through the spectrometer and are counted. Above about 1. 50 Mev the background increases due to the increasing number of protons passing through the spectrometer until at 1. 583 Mev, the highest energy at which -18 data were taken, the background was 50 counts per integration and the alpha counts were 160 - 50 .. 110 counts per integration. The protons did make larger pulses on the average than the alpha particles as was indicated by the large number of counts in the surplus channeL However, the low energy tail of the proton distribution was enough to give 50 counts in the alpha-particle region. At the peaks of the narrow isolated resonances the counting rate ranged from a low of 80 counts per integration at the 825-kev resonance to a high of 290 counts per integration at the 747-kev resonance. An example of the raw data taken at the 672-kev resonance is shown in Fig. 4. The result of each integration is shown on this graph . In the data analysis the points at each energy were averaged to- gether. The highest counting rate was observed at the peak of the broad 1274-kev resonance and amounted to 630 counts per integration. It will be recalled that there is a considerable amount of 0 in the nickel oxide targets (Section III-A). for the 018(p, a )N 15 18 Even though the Q-value reaction is about 4 Mev one would suspect that occasionally some of these alpha particles would scatter through the spectrometer and be counted when the spectrometer is set to . 17 14 count the much lower energy alpha particles from the 0 (p, a)N reaction. of 0 18 To check that there actually was a considerable amount in the target a O. 3-mil aluminum foil was inserted between the target and the spectrometer entrance in order to slow down the alpha particles from the 0 by the spectrometer. 18 sufficiently to allow th e m to be bent The target was then bombarded with 8 46-kev -19protons and a target profile was taken. in the 018(p,a)N 15 It is known that a resonance cross section occurs at this energy and its width is about 50 kev (Carlson, 1961). yield was observed. As expected a large alpha-particle The resonance effect of this yield was observed by shifting the energy several tens of kev to either side of the original bombarding energy. In taking the 017(p,a)N 14 data with these nickel oxide targets the effect of the alpha particles from the 0 18 was ob- servable only when one attempted to measure rather low, off resonant cross sections. in these cases. For this reason these targets were not used It is reasonable to assume that the 0 18 contamina- tion does not significantly affect the data taken at the narrow resonances and at the higher yield broad resonances since, a) the alpha particles from the 0 18 could hardly be expected to show sharp reso- nance effects and b) the ratio of the alpha-particle counts found with targets made with the Israel gas to that found with targets made from the Isomet gas was observed to be in the ratio of the 0 centrations in the two gases (Section III-A). If the 0 18 17 (p,a)N con- 15 reaction were making any significant contribution this would not be the case. Thus no corrections were made for the 0 18 contamina- tion--it being assumed that the entire background was given by the bombardInent of the targets made with natural oxygen . B. Iron Oxide Target Data The general procedure for obtaining data with this target was as follows. First a target profile was obtained at a proton bom- barding energy of 1.280 Mev where the alpha-particle yield is high. Figure 5 shows an example of such a profile. This first profile taken at the beginning of each run serves as a reference. All other -20data taken at other energies during that run will determine the cross section at those energies relative to the cross section corresponding to the 1. 280-Mev bombarding energy. After this ref- erence profile is taken the energy is shifted to some new value at which it is desired to measure the cross section. A complete target profile is then taken at this new energy, and this procedure is repeated if time permits. At the end of the run the energy is again set at 1. 280 Mev and the front edge of the target profile is observed. The shift in this front edge determines the amount of carbon deposited on the bombarded target spot during the run. In Section V -A4b it will be described how the data were corrected for this carbon build up. It is not permissible to shift the posi- tion of this target during the run as it was for the nickel oxide target since here the 0 sition. 17 concentration changes with target po- Because of the length of the runs necessary in the cases of the two lowest measured cross sections (at bombarding energies 804 and 854 kev) it was not possible to take all of the data at the same target spot. In this case for the different runs target spots were chosen which gave similar profiles at 1. 280 Mev, although they may have been shifted relative to one another by several millivolts fluxmeter setting. Such a low yield profile which is a combination of three runs is shown in Fig. 6. This was taken at a bombarding energy of 854 kev and corresponds to the lowest cros s section measured in the present investigation . An interesting background problem was encountered with this target. Figure 7, a target profile at 1. ODS-Mev bombarding energy, illustrates this nicely. At low-yield points it was dis- covered that the target profiles do not go to zero counts when -21. 17 14 the fluxmeter setting is such that no alpha particles from 0 (P. a)N should be observed. Instead it appeared as though there were a con- tinuUIIl of alpha particles underlying the 017(p.a)N 7 this a.I'I)ounts to about 15 counts. 14 peak. In Fig. These counts are not due to elas- tic ally scattered protons since only two or three counts at most were observed from bombardment of a piece of stainless steel under the same circUIIlstances. It did not seem reasonable that these counts could be coming from any elements (N. C. 0) known to have been deposited on this target in the mass separator (Ahnlund. 1957). The . . . d f or t h·IS target In . th e manner 0 18( p. a )N15 reaction was Investigate discussed in Section IV -A and the 0 18 concentration was found to be much too low to explain the background. Surface contamination with F19 was another possibility; although the bombarding energies used for the iron oxide data were carefully chosen to avoid the known F 19 (P. ay)O 16 resonances. A target profile was taken at a bombard- ing ~nergy of 877 kev and the alpha-particle peak from the F19 contamination was observed; however. the continuum background was also observed at fluxmeter settings where no alpha particles from F 19 should be counted. It was finally decided that perhaps there was some contaminant in the stainless steel backing (presumably made in Sweden) that was not present in the stainless steel which was being used to check for scattered proton background . The back of the tar- get blank was faced off on a lathe. then polished and cleaned. It was then exposed to the proton beam and a profile was taken. The results. shown in Fig. 7. indicate that the observed continuum was being produced in the body of the stainle s s steel backing. All back- ground data for this target were then taken by bombarding the back . -22surface of the target. The target holder was arranged so that ~he front and back of the target could be bombarded alternately simply by rotating the target through 180 0 • In this way the same target spot was always brought back into the beam. The background points taken in this way are also indicated on Figs. 5 and 6. No attempt was made to determine the actual reaction causing this background. The 10-channel analyzer was also used in taking the target profiles of the iron oxide target. Figures 8 and 9 are examples of pulse height spectra taken at energies where the alpha-particle yield was relatively low. In addition to the three integrating capacitors discussed in Section IV -A two other capacitor combinations were made use of in taking the iron oxide data. The capacitance of these was deter- mined in the same manner as before. -23V. A. ANALYSIS OF EXPERIMENTAL DATA Conversion of Data to Cross Section vs Energy 1. The Energy Relation If one neglects scattering and straggling effects then the follow- ing description of the events leading to the production of part icles which pass through the spectrometer is valid. A particle passing through the electrostatic analyzer with energy E target w ith energy E 1B e , the bombarding energy. is incident on the It moves into the target, losing energy as it goes, until its energy is E , the reaction 1 energy. The particle then initiates a nuclear reaction producing the particle to be observed with an energy E . 2 This particle then emerges from the target with energy E trometer with energy E m . and proceeds into the spec20 17 14 For the 0 (p,a)N reaction in the present experiment we have (11) (12) where V is the target potential above ground. t For the case where the target normal bisects the angle between the incoming particle direction and the outgoing particle direction the following formula may be derived* (Brown, 1951) (13 ) £1 *The work of Bardm (1961) yields a formula which dillers somewhat from Eq. (13). Near the target surface, however, they are essentially the same and so Eq. (13) will be used here. -24In this formula BE/BE 1 and E2B may be found from kinematics, E2B being the energy of the outgoing particle when the incoming particle had reaction energy E 1B . The quantities £ 1 and £ 2 are the stopping cross sections.in the target material for the incoming and 'Outgoing particles respectively. In the present analysis £ 1 was evaluated at the energy EIB and £ 2 was evaluated at the energy E 20 . Equation (10) was used to determine the stopping cross sec- tion in nickel oxide for both the protons and alpha particles. Data on stopping cross sections of alpha particles is sparse and Whaling's (1958) compilation gives only proton values for nickel and oxygen. The alpha-particle stopping cross section £ 2(E ) at enera gy Ea was computed from the proton stopping cross section £ 1 (Ep) at energy E p by use of the foHowing relation: (14) Here a is a factor that ranged from 3 .7 to 4.0 in the present experiment and is tabulated by Whaling (1958) as a function of alphaparticle energy. For the nickel oxide targets the procedure to obtain the reaction energy El was as follows. From the electrostatic analyzer setting Eqs . (1) and (H) may be used to obtain E IB . From the mag- netic spectrometer setting Eqs. (2) and (12) may be used to obtain E 20 . Equation (13) is then used to calculate E . l It is this energy that is shown in Fig. H and that has been converted to the center of mass (c.m.) energy shown in Fig. 22. The penetration depth into the target before reaction is EIB - El and ranged from about 4 to 10 kev in the nickel oxide target data. This depth was always such -25that the entire lamina £1 observed by the spectrometer was inside the oxided region of the target . The reaction lamina thickness to the protons in energy units is given by (15 ) where R =piLlp is the resolution in momentum of the spectrometer (Brown, 1951). In the present work £1 was in the range of 3 to 4 kev. It will be recalled that for the iron oxide target no shifting of the target during a run was possible; therefore the determination of a reaction energy was complicated somewhat by the carbon contamination on the target surface at the beginning of a run and by the continuing build up of carbon during the run. energy for this case was computed as follows. The mean reaction Let primed symbols refer to those quantities to be evaluated at the reference bombarding energy of 1. 280 Mev, and let unprimed symbols denote quantities I to be evaluated at the other energies . The proton energy ElM cor- responding to the midpoint on the front edge of the target profile at I the beginning of the run and the target thickness £t in energy units may be found from the reference profile by use of Eq . (13). The I initial carbon thickness £c is then (16) I I After the run the midpoint has shifted to a new value ElM + LlElM I where the energy shift LlElM is given by I I I LlElM = ElM (final) - ElM (initial) (17) These quantities may be converted to other energies by the relations -z6, £NiO ~' AEIM ~t = , "'t £NiO (18) where £ C stands for the proton stopping cross section in carbon, and the stopping cross section for nickel oxide has been used in the oxide region. This was done because the exact target composition is not known. Only a small error will be introduced since it is only the ratio of the stopping cross sections at different energies that is involved. After Eq. (18) was applied the reaction energy El was calculated by using (19) z. Charge Exchange in the Target The magnetic spectrometer was set to count the doubly charged alpha particles emerging from the target. Before emerging from the target surface the alpha particles produced by a reaction in the target undergo a sufficient n=ber of collisions to reach charge equilibri= ,. In Allison's (1958) notation the equilibrium fraction of H e ++ , He+ , and He are denoted by F Zoo' F loo ' and F 000 respectively. The values for these ratios are taken from 'Allison (1958) and are shown in Fig. 10. The alpha-particle energies in this experiment are in the range 1 to Z Mev. To obtain the true number of alpha particles produced in the reaction the number that pass through the spectrometer must be divided by F Z00 . ' -27- 3. a. Thick Target Formula General discussion. The laboratory yield Y L will be de- fined here as the number of reactions (or scatterings) produced per incident particle per steradian which, in the absence of charge exchange in the target, would be detected in a given experimental arrangement. If angle effects are neglected the yield may be re- lated to the laboratory cross section CJ"L per unit solid angle through YL = S :L f(E) dE 1 (20) where €l is the stopping cross section per target nucleus for the incident particle, and f(E) is an instrumental resolution function. Target nucleus means that nuclear type which is producing the reaction or scattering under observation. The thick target assumption is that f(E) for the spectrometer cuts the integral off before any depletion in target atoms (€l- (0) does. lar with length ~l as given by Eq. (15). f(E) is taken to be rectanguOn the assumption that the cross section is constant over the range of the spectrometer lamina (21) where CJ"L has been written CJ"L(OL) to emphasize that it is the cross section per unit solid angle . The yield is measured by bombarding the target at energy ElB with the spectrometer set at Em until a charge Q has been deposited on the target and NO reactions have been observed . The yield is then given by (22) -28where e is the electron charge, [2 is the spectrometer solid angle and N, the true number of reactions produced is related to NO' the observed number, by means of a charge exchange correction. The energy corresponding to the cross section given by Eq. (21) is the reaction energy E , Eq. (13). l b. Measurement of spectrometer solid angle . It is seen from Eqs. (21) and (22) that the ratio [2 / R for the spectrometer must b e known in order to obtain the cross section. This ratio was mea- sured by observing the Rutherford scattering of protons on a thick, evaporated eu target. The cross section can be calculated from the Rutherford formula and the number of counts NO at the peak of the target profile then yields the ratio [2/ R . Several sets of data were obtained at ElB ;:: 1. 005 and 1. 609 Mev and ~o correction for charge exchange was made. The result is [2/ R ;:: (4.03 + 0.14) x 10 -5 steradians From the slope of the front edge of the target profile R was found to be given by R ;:: 107. 3 + 1. 5 yielding [2;:: 4.31 + 0.17 millisteradian This is about 30 % lower than the value given by Snyder et a1. (1950) due to the fact that baffles have been inserted into the spectrometer since their paper was published. -29- c. Application to nickel oxide target data. For the nickel oxide data the charge exchange correction gives N = NolF 200 and €l 0.0397 gives the stopping cross section per 0 17 atom. = €NiOI Equations (21) and (22) may then be used to obtain the cross section ITL vs the reaction energy E . l In Fig. 11 is shown the results of analyzing the data in this manner. No corrections have been made for instrumental resolution·. The width of the narrow resonances below 1. 2 Mev is almost entirely due to instrumental resolution and energy straggling (see Section V -C4 below) and thus the ordinate is really not the true cross section for these resonances. The actual cross section is much narrower and rises to a higher maximum value. Figures 12 to 18 show the data from the seven narrowest resonances plotted as alpha-particle yield vs proton bombarding energy at the target surface. The dashed curves on these figures are calculated curves taking into account straggling and the spectrometer resolution. They will be discussed in Section V -C4 below. 4. a. Thin Target Formula General discussion. In deriving the thin target formula one assumes that the entire target contributes to the yield. feE) is set equal to one and the cross section is assumed not to vary over the target thickness. Thus from Eq. (20) we have (23 ) where nt is the number of target atoms per cm coming particle beam. 2 as seen by the in- Y L is now the yield from the entir~ target and Eq. (22) is not valid in this case. To obtain the yield from the -30entire target with a magnetic spectrometer a complete target profile NO vs V m must be taken. eR YL=ITQ The yield is then given by S N eR y-dVm=QQ m S V NO F dV m Zoo m (Z4) Provided that nt is known the cross section may be found from Eqs. (Z3) and (Z4). b. Application to iron oxide target data. From Eqs. (Z3) and (Z4) we see tha: the ratio of the cross section (1'"1 at an energy El to the cross section "(I'"Z at an energy E Z may be written (Z5) as long as the same target spot is bombarded so that nt is the same at both energies. By using Eq. (Z5) the cross section at any energy may be measured in terms of the reference at EIB = 1. Z80 Mev. All the cross sections are then normalized by normalizing the reference to the absolute measurements made with the nickel oxide data. In order to evaluate the integral appearing in Eq. (Z5) the quan- tity NO/ V m F Z00 was plotted vs V find the area under the curve. m and a planimeter was used to Such a plot is shown in Fig. 19--this being derived from the data shown in Fig. 5. " When significant carbon build up was observed the V N/ V L:.V V m m m spectrum was plotted. m -value was corrected before the If, at the end of a run, a total shift was observed (see Section IV-B) then, before plotting, each -value was shifted an amount fL:.V m where f is the ratio of the -31- charge deposited up to and including the setting V charge deposited during the entire profile . zn to the total This aznounts to as suzn- ing that the carbon thickness is directly proportional to the aznount of charge deposited on the target. This procedure is valid as long as the cross section does not vary rapidly w i th energy in the region where the profile is taken. In the two cases (see Section IV -B) where znore than one reference profile was taken a correction for relative shifts of these reference profiles was also applied. In Fig . 20 is shown the 854-kev data after such corrections are applied. The uncorrected data for this figure is shown in Fig. 6. Cross section values zneasured with the iron oxide target are shown, along with nickel oxide values, in Figs. 11 and 22. In order to znore clearly display the lowest cross section values which were zneasured a sezni-log presentation of the data between 660 and 1100 kev is given in Fig . 21. In this figure the positions of the n a rrow resonance s are indicated by vertical lines. B. Error Analysis The statistical error for all the data below El :: 1. 450 Mev is clearly i ndicated in Figs. 22, 21 , and 12 through 18. Mev the statistical error was about 60/0' Above 1. 450 Table I shows the errors in the quantities needed to obtain the error in the a bsolute cross section (error shall znean the rzns deviation). errors in the quantities used to obtain CT L Compounding the results in an error of 10% to be coznpounded with the statistical error to yield the error in the absolute cross section. The bulk of this error is seen to arise frozn the uncertainty in the alpha-particle stoppi ng cros s section E 2. The following discussion relates in znore detail how one -32arrives at the errors quoted in Table 1. F 200: No errors in this quantity are quoted by Allison (1958); however, the bulk of the data is taken from Dissanaike (1953) . quotes an error of 2% in He ++ / He+. He In the range of alpha-particle energies dealt with in this experiment this leads to a 2.60/0 error in F 200. 17 . 0 .conc entr ahon: The manufacturer of the enriched gas gave no error in this quantity. One percent is a reasonable guess since the concentration was quoted to three figures and this would give an error of a few in the last figure. Q: The error here is composed of the error in the capacitor measurements and the error in the measurement of the mercury batteries. n / R: These errors are small and lead to a 0.4% error in Q. The main contributions to this error are from the Cu stopping cross section (4%) and in the value of the integrating capacitor (2%). 0.007 fLf. A small capacitor was used here having C :at 0.319 + This gives a 3.6% error in n / R. Kinematics: The kinematic relations are based on the general principles of conservation of momentum and energy and are assumed to hold exactly. E m : The 0.20/0 error in this quantity comes from the error in the spectrometer constant k m and an assumed 1 part in 1000 long- term stability (Milne, 1953). E (p in NiO): In assuming a 4% error in the values for E Ni and EO as read from Whaling's (1958) graphs one obtains the quoted 3.0% error. Eq . (4)]. No error was assigned to the assumption fL =1 [see -33E (O! in NiO) : In Whaling ' s compilation (1958) in whi ch he tabu- l a tes the quantity ~ defined in Eq. (14) he assigns a 20'70 error to this quantity. It is felt however that in the range where a is near 4 this is an overestimate and that something of the order of 10'70 would not be unreasonable. Thus a 10'70 error was assumed in the values used for the stopping cross section of alpha particles in nickel oxide. E (8Ez!8E ) + E 2 : This quantity is quoted in Table I since it l l is this combination of stopping cross sections which appears in the cross section formula. Using the above adopted errors for the stopping cross sections it was found that the error in E (8E / 8E )+ E 2 2 l l was very close to 9'70 over the range of energies employed in this experiment . Compounding these errors then gives a 10'70 error in the scale of absolute cross sections. Since the iron oxide data was normalized to the nickel oxide data this 10'70 factor holds for data taken with both targets. We now discuss the errors to be assigned to the energy scale. The relation between the reaction energy and the bombarding energy may be written (26) where the correction term E E or by c c is given either by nickel oxide (27) -34E (28) iron oxi de c For the nickel oxide target data E c was in the range of 4 to 10 kev with a percentage error of about 12 0/0. An error of 0 . 2% is assigned to E1B due to the error in calibrating the electrostatic analyzer and to account for long-term drifts in the equipment. No detailed investigation of long-term drifts was undertaken; however the consistency of the 0 17 14 . (p, a)N and Cu(p,p)Cu data over a long penod testifies to the smallness of this effect. in an uncertainty of about 0.25 % in E . l These errors then result Since this is of the same order as the electrostatic analyzer resolution it is felt that an increase of the assigned error to 1/ 3% is reasonable. For the iron oxide target data Ec was in the range of 10 to 20 kev with an error of about 9 % . This again leads to an error of 1/ 3% in E . l In summary we have Reaction energies accurate to 0.33% Absolute cross section scale accurate to 10% C. Extraction of Nuclear Parameters 1. Barrier Factors The ratio of the partial width r X.c for break up of a compound state into channel c via level X. to the corresponding dimensionless reduced width 9~c will be called the barrier factor * . The nuclear 2 effects are reflected in the quantity 9x.c; the barrier factor takes *This barrier factor is not the same as the various penetration and transmission factors used in the current literature. -35into account the Coulomb and centrifugal barrier and may be calculated. The barrier factors for the 0 17 (P. a )~4 reaction have been calculated for several values of relative orbital angular momentum near the observed resonance energies and also at several very low energies. The calculations are described in Appendix A and the re- sults are given in Tables II and Ill. Several of these barrier factors will be needed in the analysis to follow. 2. Angular Distribution Factors No angular distributions were measured in this experiment. All of the data were taken at a laboratory angle of 150 angle =152 0 ). o (6 = c.m. It is of interest therefore to obtain some idea of the error that might be involved in computing the integrated cross section simply by multiplying the cross section per unit solid angle at 6 =152 by 471". An appropriate factor {3 which will measure the 0 error in this assumption is (3 = 471"0-(152 o ) (29) S 0-(6) <ill and {3 will be unity when no error is made. In fact to obtain the integrated cross s"ection one should multiply the measured differential cross section by 4 71"/ {3. of 0 17 is 5/ 2+ and that of N 14 The ground state spin and parity is 1+ (Ajzenberg-Selove. 1959). Thus in terms of angular momentum we may write 1 J71" __1_-,>~ 1+ + 0+ where 1 is the relative orbital angular momentum in the 0 17 system. 1 1 is the relative orbital angular momentum in the N +P 14 +a -36system, and J 7r i s the total angular momentum and parity of the state in the compound nucleus FIB formed in the reaction. The in- coming channel spin s can be e ither 3+ or 2+ and the outgoing channel spin s' is 1+ In general the factor {3 depends on the channel spin mixture assumed in the incoming channel. Limits can be put on {3 , however, and these limits are given in Table IV. The quan- = 2 and {33 corresponds to pure s = 3. tity {32 corresponds to pure s The actual {3 must lie somewhere between these two values . Cal- culations were not made for some of the higher 1, l ' value s for a Note that a 0+ level is completely forbidden to decay into N 14 + a. The calculation of {3 is discussed in Appendix B . The results in Table IV indicate that in a few cases an error by as much as a factor of 2 would result from the assumption that {3 =1. Thus if the spin and parity of a level are not known one would expect a maximum error of about a factor of 2 in assuming {3 '" 1 but could reasonably expect this error to be less. 3. Broad Resonance Analysis The data in the region El = 1. 0 to 1.'.5 Mev (Fig. 11) h a ve . . . . th e 0 17 + p s y s t e m. b een converted t o c. m. quanhhes ln results are plotted in Fig . 22 . Here we describe how the resonance energies and widths of the high-yield broad resonance near E kev and of the low-yield broad resonance near E determined. The =1200 =1040 kev were In thi s section the effects of the narrower anomalies at c.m. energies of 1040, 1177, a nd 1260 kev are neglected. The spin and parities of these broad levels are known to be 2- a nd 1 for the high-energy and low-energy levels respectively and both -37are known to decay by p-wave alpha particles (Ajzenberg-Selove, 1959). It will be assumed in what follows that the protons also are p-wave. Since the levels do not have the same spin and parity the cross section integrated over solid angle would show no interference between the two levels. The cross section at a given angle however (this is all that was measured in the present experiment) will in general be expected to show interference effects, even between levels of different spin and parity (Lane, 1958). The correct general ex- pression (which would include unknown phases) will not be used here. Instead it will be assumed that a simple sum of two single level contributions will suffice to determine the resonance energies and total widths to reasonable accuracy*. the level shift will also be neglected. The energy variation of The equation used to fit the two levels under discussion was thus taken to be r pl / + same term with 1...... Z ] (30) where 1 refers to the high-energy resonance and Z refers to the lowenergy resonance. and gz Here g is the usual statistical weight, gl = 5 / 1Z =1/ 4; lC is the reduced de Broglie wavelength; f3 has been de- fined in Eq. (Z9); E r is the resonance energy; and the r s are the appropriate partial widths. The factors f3 / 411" were included in an *It is to be expected that such a procedure will give good accuracy for the high-yield resonance parameters and relatively poorer accuracy for the low-yield resonance parameters. No estimate of the effect of neglecting the interference was made however. -38attempt to obtain a reasonable normalization. The actual value of {3 which is chosen will strongly affect the values of the individual partial widths which are assumed but will have very little effect on the total widths or resonant energies. {31 The values chosen were =1. 00 and {32 = 0.933. The value for {31 based on the observed isotropy in the N 14 (a, p)017 reaction for this level in F18 (Kashy, 1958), and the value of {32 is that calculated for a pure J 1f I. =1 , - = I. ' =1 level (see Table IV and Appendix B). In using Eq. (30) the energy variation of the bar r i e r factors r / 8 2 was included. Several values for these factors in the desired energy range are given in the column labeled I. =1 in Tables II and III. Polynomials of the form 4 ~! =L anEn p n=O (31) were fitted to these points. Figure 23 shows both the points taken from Tables II and III and curves calculated from Eq. (31). The co- efficients appearing in Eq. (31) are listed in Table V, both rand E being measured in kev. Equation (31) was then used to obtain the barr,ier factors to be used in Eq. (30). widths 8 2 "best fit" and the resonance energies E The diInensionless reduced r were then varied and a was determined by inspection of a graph on which both the experiInental points and the cross section as calculated from Eq. (30) had been plotted. This "best fit" is shown as a dashed -39curve in Fig. 22. The resonance energies and widths were found to be Erl =1203 + 2 kev r l = 79 + 5 kev Er2 ::: 1035 + 5 kev r 2 ::: 85 + 5 kev where the errors are estimated from the analysis and do not include the errors in the energy scale as discussed in Section V -B. The energy scale errors are, however, included in Table VII, which gives a summary of all the level parameters as determined in this experiment. The partial widths are not uniquely determined from the data. Even if one assumes that Eq. (30) holds exactly there is still an ambiguity as to whether or not it is the proton width or the alphaparticle width which is the largest. Table VI gives the partial widths which are consistent with Eq. (30) and the data. Group I are the values used to· calculate the curve on Fig. 22 . Group II could also have been used, however, as could a combination of Ia with lIb or Ib with IIa. This would result in only slight changes in the calculated cross section. No analysis was performed on the resonance at E = 1260 kev. The other two narrow levels appearing on Fig. 22 are discussed in the next section. 4. Narrow Resonance Analysis In Figs. 12 to 18 the alpha-particle yield from the narrow reso- nances is plotted vs proton bombarding energy. Most of these graphs have a full width at half maximum of about 5 kev . It is the purpose -40of this section to determine the source of this width and to extract from the data as much information as possible about the level parameters for these resonances. Three sources of experimental width will be considered: magnetic spectrometer energy resolution, energy variation with angle, and energy straggling. The thickness in energy units of the target lamina to the incoming protons due to spectrometer resolution is about 3.5 kev as calculated from Eq. (15). listed as Sl (The actual thickness at each resonance is in Table IX.) It is clear then that additional contri- butions to the observed energy width are present. Since the full spectrometer solid angle was used in the experiment it was decided to investigate the question of the energy spread introduced due to energy variations with reaction angle. reaction angle was SLab =150 0 , The mean however the actual reaction angle can vary somewhat about this mean due to the finite spectrometer acceptance angle. The formula for the energy width in a yield curve produced by a zero width resonance and observed by a spectrometer with infinite energy resolution but with an angular acceptance angle Ii 0 has been c;alculated in Appendix C. The work of Bardin (1961) may also be used to derive the result. The resulting energy spread for the situation in this experiment is given in Table IX under the column labeled .6. . 0 These are all small--of the order of 1 kev-- and would be expected to have only a small effect on the observed width. In fact if two rectangular distributions of widths 3.5 kev and 1. 0 kev are folded together the result still has a full width at half maximum of 3.5 kev. It is quite reasonable to assume rectangular distributions for the spectrometer energy and angular resolution. -41Finally we consider the effect of energy straggling on the observed y i eld width. Both the incoming protons and outgoing alpha particles will suffer from this. In Appendix D a formula is derived for the laboratory alpha-particle yield Y L (E lB ) as a function of the proton bombarding energy EIB for the case of a very narrow (6function) resona nce in the cross section. In this formula the spec- trometer energy resolution and particle straggling are taken into account [ see Eq. (D.ll)) . Angular effects are neglected. The Bur- roughs 220 computer was programmed to perform the necessary integral--the parameters used at each resonance are given in Table VIII. The resonance energies used in the calculation which were found to give the correct position of the peak of the yield were the same as those given by Eq. (13) to within a few tenths of a kev. The results of these calculations are shown as dashed curves in Figs. 12 to 18. It appears that in most cases the calculated energy spread is enough to account for the observed width. This would then imply a rather small width for the resonance involved. For the l247-kev (Lab) resonance (Fig. 18). however, there is a definite contribution to the observed width from the natural width of the resonance and for the 747-kev (Lab) resonance (Fig. 14) there seems to be a much smaller , but observable , contribution. One can notice a discrepancy between the calculated curves and the experimental data which occurs in the w i ngs of the yield curves . In particular the high-energy experimental points lie higher than the calculated curve. This is due to the fact that Gaussian dis- tributions were used for the straggling functions and can be explained -42qualitatively in the following way. The actual straggling function is not Gauss i an but has a higher low-energy tail than a Gaussian. Thus, at a high bombarding energy , more protons than calculated will slow down to the resonance energy by the time they reach the target lamina in which the m a in contribution to the spectrometer y i eld is produced. This would make the high-energy counts greater than those calculated--as observed. A quantitative explanation would require use of the correct straggling functions (Rossi, 1952). It is quite easy to obtain the resonance energies from the data but very difficult to extract the widths with any degree of accuracy. The area under the yield curves, however, can be measured accurately and this area can be related to the resonance parameters. It is shown in Appendix D that for a a-function cross section and Gaussian straggling functions the area under the yield curve is related to the area under the true cros s section by the same formula one obtains if straggling is left out and a nonzero value is assumed for the natural width, r. It seems reasonable to assume that if the proper straggling functions are used and the cross section is allowed to have a small width then the same conclusion would hold. We then may write [see Appendix D Eqs. (D.2l), (D. 26)] 4 S (>.7r t-' where E m 2E YL(E lB ) dElB =Re (max) (;n Ie + 0') (mTrr/ 2)L~ s lr 2r lr (32) (max) is the spectrometer energy setting at the maximum of the yield curve, ~ is defined in Eq. (29), Y L is the yield in alpha particles per proton-steradian as shown in Figs. 12 to 18, and 0'= 8E m / 8E lB . The subscript r on the stopping cross sections denotes that they are to be evaluated at the resonance energy a nd the quantity -43(IT r 7r/ 2)L is defined in Appendix D and is simply the total area r under a single, narrow resonance in the laboratory system of coordinates. We may then calculate the quantity f3g r a r p I r in the c . m. system by means of the following formula f3g r r 0L 2MMOE El R(E 2 l EI + --J.--P= (--) r r r r (33 ) °CM -fl2(~ + MO)Em (max) with the statistical factor g given by g = 2J12+ 1 (34) where J is the spin of the appropriate state in F 18 , E r is the reso- nance energy in the c. m. system, and 0L/ oCM is the solid angle ratio as computed from kinematics. This ratio varies from reso- nance to resonance but is asswned not to vary over a single narrow resonance. M is the reduced mass in the 0 and MO are the proton and 0 17 17 + p channel, and ~ mass respectively. The results obtained from applying Eq. (33) to the several narrow resonances are given in Table VII. (33) was evaluated with a planimeter. The integral involved in Eq. In Figs. 17 and 18 the dotted curves were used as the base line in evaluating the integrals. In Table VII are also quoted estimates of the total widths. These were obtained by asswning that the spectrometer-straggling width, the small width due to the angular aperture of the spectrometer, a n d the resonance width combine as the square root of the swns of the squares to yield the measured width. The measured width was read from the graphs, and the spectrometer-straggling width was read from th e dashed curve s . -44In between the narrow isolated resonances where the cross section was quite low the data allow an upper limit to be placed on the quantity I3g I3g ra rp/ r r r a p <8 ev r for any unobserved narrow resonances. unobserved narrow resonances In fact below i280 kev (Lab) the data were taken at small enough energy intervals so that this upper limit on unobserved narrow resonances is applicable from 490 to i280 kev. Above i280 kev the data were taken in larger energy steps so it is possible that in the range from i280 to i580 kev some narrow resonances with I3g r a rp/r exceeding the above limit are present. It should be emphasized that even though actual cross section measurements are not shown in Fig. ii in the region of very low cross sections, data were taken in these regions (for example, the off-resonance data shown in Fig. 4) which give the upper limit of 8 ev quoted above. D. Discussion of Experimental Results In the subsequent discussion all quoted excitation energies in F i8 are based on the following mass differences (Everling, i960) Oi7 + Hi _ F i8 :;; 5.597 Mev N i4 + He 4 _ F i8 '" 4.404 Mev All the levels observed in the present work have been previously reported except possibly the 747-kev level (E (Ajzenberg-Selove, i959). ex = 6302 kev) Hinds and Middleton (i959) in a study of Fi9(He3,a)Fi8 report a level at 6264 kev excitation . This could ~45- possibly be the present 747-kev level since in their data the alphaparticle group for thi s level is not completely resolved from the group (about three times stronger) leading to the 6.232 level in F18. The upper limits found here for the widths of the narrow levels (Table VII) are all consistent with previously reported limits. Width limits for the levels at E previously been reported. ex = 6376 and 6472 kev have not Also, the width of the level at E ex :;; 6774 kev had not previously been measured. There have been several discrepancies in the literature on the reported widths of the broad levels at E (Fig. 22). ex = 6632 and 6800 kev The results of these previous measurements along with those of the present work are given in Table XI. these discrepancies may easily be explained. Several of It is clear from the papers of Ahnlund (1957), and Heydenburg and Temmer (1953) that their quoted widths for the 6800-kev level include the contribution from the higher energy 6857-kev level. It is fairly certain that this is also the case for the value quoted by Herring (1958b). In the work of Kashy et al. (1958) it is not clear whether or not their quoted value for the width of the 6800-kev level includes this extra contribution. kev. They do not report a level at E ex = 6857 In any case their value for this width is in agreement with the present work. If the 6857-kev level contribution were added to the value obtai ned in the present work then a value of about 100 kev would be obtained (Fig. 22) in agreement with Heydenburg and Temmer (1953) and with Herring (1958b). The situation for -46the 6632-kev level is somewhat puzzling. The value of 27 kev (Heydenburg, 1953) may be discarded since it is clear that the narrow level at E ex = 6637 kev strongly influenced the data from which this width was derived. The present work agrees with the work of Herring (1958b) and not with Kashy et al. (1958). This seems odd since the situation was simply reversed for the 6800kev level. No completely satisfactory explanation has been found for this situation . Kashy et al. carried out a detailed analysis, but only quote the resulting parameters; whereas Herring shows the fit which he obtained for the two broad resonances in question. at E ex Kashy et al. do not report the narrow level = 6637 kev whereas Herring does--this suggests that this level may possibly have influenced the results of Kashy et al. It is felt safe to say that the present work is in agreement with that of Herring (1958b.) provided that his width for the 6800-kev level is taken to include the contribution from the 6857-kev level. In the region of excitation of F18 under investigation in the present experiment two levels have been reported in N that were not seen in the present work. l4 (a, a)N The se are a level at = 6247 kev (Herring, 1958a, 1958b; Silverstein, 1960) and one at E = 6556 kev (Herring, 1958a, 1958b). The 6247-kev ex E ex level is known to be formed by s-wave alpha particles making J 7T = 1+ . Thi s would require d-wave or g -wave protons to form the state. 2 Since Table II shows that r/ e drops by a factor of 20 on going from p-wave to d-wave protons the absence of this level in the present experiment could perhaps be explained on l4 -47the basis of these angular momentum considerations. The 6556- kev level is thought to be formed through g-wave alpha particles giving J'Ir = 3+, 4+ , or 5+ for the level. It is felt that the absence of this level in the present experiment is evidence for discarding the 3+ possibility since a 3+ level can be formed by s-wave protons whereas 4+ and 5+ levels require protons of d-wave or higher. From the quantities I3grarp/ rlisted in Table VII one can, if so desired, obtain a rough estimate of the minimum partial width involved at the level in question. r Due to the fact that = r a + r p (we neglect the radiative width) one can show that where r , is the minimum of r , r . mm a p The equality on the left holds when one of the widths is much smaller than the other , 1. e . when r ' «r = r , + r • mm mm max The equality on the right holds when the two widths are equal, 1. e. when r This then gives r ' mm = r max = r / 2. , to within a factor of 2 (as suming that 13 and mln g are known). E. Summary of Experimental Results To summarize we note that Fig. 11 shows the complete excitation curve for the 017(p,a)N present work. 14 reaction as measured in the Figures 21 and 22 show selected regions of this curve and Figs. 12 to 18 show the alpha-particle yield curves at the narrow resonances. Table VII summarizes the level param- eters as determined from this investigation and Table VI lists -48a consistent set of partial widths for the two broad resonances. Any unobserved narrow resonances below E1 j3gra r/r < 8 ev. =1280 kev have -49VI. THE ASTROPHYSICAL PROBLEM A. General Discussion In a recent paper Fowler, Greenstein, and Hoyle (1961; re- ferred to as FGH) propose that the relative abundance of several of the elements which were present in the original material from which the solar system condensed was modified by spallation processes and neutron interactions during the formation of the planets. Among their conclusions is that the 0 17 abundance was not modified appreciably by these processes and that therefore . 17 16 the ratiO 0 /0 present on the earth should be the same as for the primitive material. We use the symbols of the elements to stand for their relative abundances or the number of nuclei 3 per cm , depending on the context. Following the notation of ~ stand f or t h e orlglIla . . 1 pro d uctlon · . 0 f 0 16 re 1aFGH we 1et t-' ratlo tive to C 12 in helium burning in red giant stars, and we let f represent the fraction of the C 12 and 0 16 thus formed which has been processed to equilibrium in the CNO-cycle. 0 17 If we add the ratio to an equation given by FGH we find C12:N14:016:017 = (1 - f + O. 024f(1 + 13)] : o. 95f(1+I3) (13(1 - f) + 0.02£(1 + 13)] : '(f(1 + 13) (36) where the element symbols refer to the concentrations in the primitive material. This relation assumes that in the CNO-cycle the equilibrium abundance ratios are given by (37) -50The nwnbers in Eq. (37) come from asswning equilibriwn in the CNO-cycle at a temperature of 35 x 10 6 0 K. This rather high temperature comes from the asswnption that the last CNO-cycle processing undergone by the 0 16 and C 12 which were destined for the solar system took place in stars which were at a rather advanced stage of evolution. In these stars the hydrogen burn- ing occurs in a thin high-temperature shell surrounding a helium core . 0 17 /0 ture. We shall see below that the present calculation of the 16 ratio does not seem to agree with such a high tempera- FGH (1961) next asswne that the present solar abundance ratio C12:N14:016 ;:; 5.5 : 1 : 9.6 represents the original ratio of these elements in the primitive material. This asswnption, along with Eq. (36) then leads to the values 13 ;:; 1. 75 f '"' 1/15 From Eq. (36) one then finds yf(1 + 13) (38) 13(1 - £} + 0.02(1 + l3)f '"' o. Hy The observed terrestrial ratio is 0 17 /0 16 : 3.74 x 10- 4 (Nier, 1950) which leads to a value y:3.4x10 -3 The ratio at equilibriwn in the CNO-cycle is then given by Eq. (37) to be -51- [0 17 /0 16 (39) ]CNO=y/ 0 . 02=0.17 Here we assume that the only possible way of forming 0 the CNO-cycle . 17 is through This is a rather high value for this ratio and would result from a rather low rate for the 0 17 -destroying reaction, 017(p,a)N 14 . In the remainder of this section we derive an expression for the low energy cross section for the 017(p,a)N to calculate the ratio [ 0 17 /0 16 14 reaction and use this . ] CNO as a functlOn of temperature. We then compare the results with the prediction of Eq. (39). B. Cross Section Formula The observation of the very narrow levels at low energy in the present experiment leads one to suppose that these levels will have very little effect at energies of interest in stellar reactions (around 30-kev c . m. energy in the 0 . . . F 18 ) • excltatlon ln 17 + p system or 5627-kev One is then interested only in the effect of F18 levels which occur near this stellar energy region. A look at the F18 level scheme (Ajzenberg-Selove, 1959) indicates that only two of the known levels might be of importance in determining the . 17 14 stellar cross sectlOn for 0 (p,a)N . at E ex These two levels occur = 5594 and 5662 kev (using the mass differences quoted in Section V -D). The 5594-kev level has been investigated by means of the N N 14 14 (a,y)F 18 (a, a)N 14 (Price, 1955; Phillips, 1958, Alrnqvist, 1958) and (Silverstein, 1960) reactions. All results point to -52a spin and parity assignment of 1 for this level. Silverstein (1960) reports that the alpha parti cles are p-wave and that the total width is about 200 ev. The 5662-kev level has also been investigated by means of the N 14 (a.'Y)F 18 (Price. 1955 ; Phillips. 1958) and N 14 (a.a)N 14 (Silverstein. 1960) reactions . The evidence here also points to a 1- assignment for the level. Silverstein (1960) again quotes a width of about 200 ev and assigns p-wave alpha particles to the lev el. In order to calculate the 0 17 (P. a)N 14 . cross sectlOn at stellar energies we shall consider only the contribution of these two levels. Since they both have the same spin and parity the total cross section IT will exhibit interference between the two levels. We let 1 refer to the 5662-kev level and 2 refer to the 5594-kev level. The total cross section may then be written (Lane. 1958) IT 2 =1rX g (r r )1 / 2 pI al . 2 + same term with 1 -- 2 (Erl-E) f t 2 (40) ir where the energy dependence of the level shift has been neglected and the approximation that the level widths r 1 and r 2 are much less than the level spacing Er2 - Er1 has been made. imation is quite good in the present case. This approx- We have here defined (rp r a )1/ 2 to be positive and are allowing the +- in Eq. (40) to take care of the interference effects. For the 0 17 For J = 1 we have g :; 1/ 4. + P channel the energy E will be sufficiently low so that the approximation for the partial width given in Appendix A will be valid [see Eq. (A. 15)] . The energy E + Q in the N 14 +a channel is not low enough. however . to make use of this approximation. The p-wave barrier factors have been calculated in this -53energy region and are given in Table III. A polynomial of the form (41) was fittp.d to these points. The calcl,llated pOints, along with this fitted curve are shown in Fig. 25. The coefficients c n are given in Table V. It is convenient to define a quantity S(E), the cross section factor , by ( 42) with b given by Eq. (A.17). Combination of Eqs. (40), (A. 15). and (4Z) then gives (43) with SZ(E) is the same as S1 (E) with 1 replaced by Z. The interference term SI (E) is given by SI(E) = s(1-a E-) Z Z 1/ z Z(9 p1 9 pZ r a1 r a2) [(Ert-E)(E-E rZ )- r 1 r zl4] 2 2 2 2 [(E-E r1 ) t r1 / 4] [(E-E rZ ) trzl4] (45) In these equations s is given by (46) The positive sign in Eq. (43) gives constructive interference between the levels and the minus sign gives destructive interference between the levels. From now on it will be assumed that only p-wave pro- tons contribute to the reaction. . 17 14 We then fmd for the 0 (p, a)N reaction (see Appendix A) a = 5.000 fermi x s = 6.543 x 10 7 kev 2 -barns -1 a L =a l =-2.. 095 x 10 - 4 kev b C. = 3.24 = 244 . 1 kev l / 2 Choice of Reduced Widths In order to obtain the cross section factor it is necessary to 2 2 have an estimate of the dimensionless reduced widths e and e for p a the two levels in question. The total width of these levels is almost certainly due to the alpha-particle width and thus ev for the levels. r a is about 200 On obtaining the barrier factor from Fig. 25 one finds that e~ would be about 0.10 for the upper level and about 0.18 for the lower level. We shall use* 0.14 which gives r l = 268 ev and r 2 = 154 ev. *For convenience in calculation, the reduced alpha-particle widths for the two levels are taken to be the same. The general shape of S(E) between the resonances is not strongly affected by such an assumption (refer to Fig. 26). -55In order to obtain an estimate of 9 2 we shall assume that the p two levels contributing to the stella r cross section have reduced widths simila r to some of those found in the present experimental work on the 017(p,a)rJ 4 cross section. We shall now give rough 2 2 estimates for the product 9 9 on the basis of the present expera p 18 irnenta1 work and other known properties of the appropriate F levels. E ex = 6086 kev : This state decays by f-wave alpha particles and ris about 200 ev (Silverstein, 1960). On the assumption of = 3 - , and p-wave protons, we find from Tables 2 2 -3 II , III, and VII that 9 9 =1. 9 x 10 . p a angular isotropy, J 1T E ex <=: 6232 kev : This state decays by f-wave alpha particles and is known to have a total width less than 800 ev (Herring, 1958b; Phillips, 1958; Silverstein, 1960) . 1£ we assume angular isotropy, = 2 - we find, from Tables II, Ill, and VII, 2 2 -3 . 2 9 9 < 1. 06 x 10 . 1£ we substitute the assumptlon 9 = 0.14 for p a a 2 2 3 the assumption r< 800 ev we find 9 9 = 0.74 x 10- . P a p-wave protons, and J 1T E ex = 6302 kev: An estimate is more difficult here. From = 0.465 +- O. 210 kev 2 . 1£ this state is similar to the previous two in that J = 3, and that f-wave a1phaparticles and p-wave protons are involved, then setting ~ = 1 gives 2 2 -3 9 9 = 4. 3 + 1. 9 x 10 . P a Table VII we have ~g r r p a The analysis of the remaining narrow levels would involve unjustified guessing as to the orbital angular momenta 1. which were involved similar to that done in the case of the 6302-kev level. Thus this type of analysis will not be continued. From Table VI we see that 9 2 a for the two broad levels is not -56very close to 0.14; therefore, we do not attempt to apply these reduced widths to the levels in the stellar energy region. On the basis of the order-of-m agnitude results from the preceding rough analysis the value will be as sumed to apply to both of the levels in the stellar energy region. The cross section is proportional to this factor and any future revisions of this quantity will result in an appropriate scale correction to the present calculations. The results of the S-factor calculations from Eqs. (43), (44), and (45) are shown in Fig. 26 for both the case of constructive and destructive interference between the resonances. D. Calculation of Reaction Rates Many authors (Burbidge, 1957 , and references therein) have discussed the calculation of stellar reaction rates, so no detailed derivations will be given here. One assumes that the reacting nuclei are at equilibrium at absolute temperature T and possess a Maxwell-Boltzmann energy d istribution. For charged particles one makes use of Eq. (42) and obtains 1/ 2 OJ h =[ M(kT) 8 3J S 0 IP(E) dE (47) where IP(E) =S(E) exp-(b/ fE + E / kT) (48) -57In these formulas X and Y represent the number of nuclei of type 3 X and Y per ern , P is the reaction rate for X + Y in reactions per 3 cm -sec, k is the Boltzmann constant, and M is the reduced mass of X and Y. For the 0 17 P 17 -H-;-1-0'1"'7 = 14 (p, a)N reaction we may write 2.008 x 10- 15 T 3 j2 00 Sa cl>(E) dE (49) 6 In this expression T 6 is the absolute temperature expressed in units of 10 6 2 OK and the integral must be in units of kev -barns. We shall always measure S(E) in kev-barns and E in kev. Two approximations are commonly made in performing the integral in Eq. (47). One is the resonant approximation (RA) in which the main energy variation in the integrand is assumed to be in S(E) (a resonance occurs at E E = E r ). The exponential is evaluated at = E r and a single level Breit- Wigner formula is used for S(E). The integral can be performed and this yields an analytic expression for the rate. The second approximation is the nonresonant appr oximation (NRA). In the NRA, S is assumed to vary slowly with energy, is evaluated at the energy EO at which the exponential is a maximum, and then is taken outside the integral sign. The exponential is then approximated by a Gaus sian, and the integration may be carried out. Formulas for the rates in these two cases have been given by Burbidge et al. (1957). For a given reaction at a given temperature one generally uses the RA to obtain the effeet on the rate of resonances in the cross section which occur near the stellar energy region and the NRA to obtain the effect on -58the rate of the cross section away from the resonances. One effect usually predominates and the rate is quoted either as "resonant" In the present case it was suspected that at or "nonresonant". some temperatures the NRA may not be a very good way to obtain the nonresonant effect since S(E) would be changing rather rapidly. Also it was clear that for the case of destructive interference the NRA would give too low a rate at temperatures (around T 6 =15) where the maximum of the exponential occurred near the minimum of S(E) . For these reasons it was decided to perform the integral in Eq. (49) graphically. In order to facilitate this graphical integration the effect of the resonance at E r = 65 kev was subtracted out. * 1£ this were not done then a plot of 4>(E) vs E would show a very high, very narrow spike at the higher temperature s. We thus write for the rate (50) where, for the case under consideration, P r is given by the RA to be (Burbidge, 1957) -'r e r (51) with *This is the level referred to as number 1 in reference to Eq. (40). The lower resonance does not produce any difficulty since the exponential goe s to zero at E O. = -59- 'rr The correction term P = 30.28 + 754 . 7/ T 6 (52) is giv en by Eq. (49) with 4l(E) replaced by c 4l (E) where c 4l (E) c = 4l(E) - 4l r (E) (53 ) The integrand 4l (E) is simply that which is used in the RA and is r given by 4l (E) r where S r s r 2/ 4 = ___r---.._-...._ exp- (b/ IE +E / kT) = S(E r ). (E _ E ) 2 + r r 2/ 4 r In the present problem E r (54) r = 65 kev and r = 0.268 kev. The function 4l (E) was then calculated as a function of E. c It was found that even this function exhibited a behavior which would make the graphical integration somewhat difficult. below E r At energies just the function 4l (E) exhibits a sharp positive spike followed c by a sharp negative spike just above E r . This is because in the present case the function 4l (E) underestimates 4l(E) below E , and r r above E r it overestimates 4l(E). This was handled by folding the negative spike back toward the low energy end and subtracting the absolute values of the ordinates. This effectively subtracts out the negative area produced by the negative spike and the resulting curve [ called 4l cf(E)] is quite smooth. The net area under 4l (E) c is equal to the area under 4l cf(E) so we have 2.008 x 10- T 3/ 2 6 15 (55 ) -60The integrand ~cf(E) is shown in Figs. 27 to 30 for two temperatures . Both the case of constructive interference and the case of destructiv e interference are shown. The dashed curve in Figs. 28 to 30 shows a portion of the positive spike produced by ~ (E) . c The negative spike is not shown. After plotting ~cf(E) in this man- ner the area was obtained with a planimeter and the rate correction P c was calculated from Eq. (55). The results of these calcula- tions are given in the second and third columns of Table XII . resonant rate P r four of this table. as calculated from Eq. (51) is shown in column Above T6 so that Pc can be neglected. P c and P r The = 25 the resonant rate predominates The total rate P 17 obtained by adding is given in columns five and six of Table XII. It is of interest to compare the exact rate correction P c to a rate Po obtained by using the formulas for the NRA (Burbidge, 1957). The expression for Po will not be written down here [it is similar i n form to that given for P 16 in Eq . (56) below], but the comparison in the form of log (Pc / PO) vs T6 is given in Fig. 32. It is seen that for the case of constructive interference Po is a good approximation to P c but for the case of destructiv e interfer- ence a considerable error would be made in using Po above a temperature of about 13 x 10 6 0 K. It is now desired to obtain the rate P . o 16 (p , y)F 17 10 order to check Eq . (39) . resonant with So = S(E ) O 16 for the reaction The reaction is non- = 5 kev-barns (Fowler, 1960). This gives 9 . 244 x 10- 15 = -'---T~Z7""'3"'--- 6 e -T (56) -61with .,..=166 . 9/ T The rate P 16 The last two columns in this table give the . 17 16 quantlty log (0 Pl/O P 17 • (57) obtained from Eqs. (56) and (57) is given in column seven of Table XII. to P 1/ 3 6 17 ). For T6 > 25, P r has been set equal This quantity is also plotted v s temperature in Fig. 3l. At equilibrium we have P 16 = P 17 ' in which case the ordinate in Fig. 31 is just the logarithm of the ratio 0 17 /0 16 in the CNO-cycle . The top horizontal dashed line corresponds to the value for this ratio given by Eq. (39). E. Discussion of Results If the case of constructive interference applies then a reduc- tion of e~e~ by a factor of 20 is necessary in order that [0 reach the value given by Eq. (39). 17 / 0 16 ] CNO The case of destructive interfer- ence does reach the value of Eq. (39). If the present assumptions about the low energy cross section for 0 17 (p, a)N 14 . reactlon made above are correct, then it appears from Fig. 31 that the terrestrial mat e rial which has been processed by the CNO-cycle underwent this proces sing at a considerably lower temperature than the 35 million degrees a ssumed in connection with Eq. (37). At this tem- perature only the 65-kev resonance will contribute to the reaction rate and the effect of the lower level can be neglected. A reduction 2 2 0f e e by a factor of over 2000 for the higher resonance would be ap required to obtain [017/ 0 l6]CNO = 0.17 at T6 = 35 . This is not impos sible, but seems unlikely in view of the experimental results -62of the present work. pute [0 17 /0 16 In order to be consistent one should now com- . ] CNO at thlS lower temperature rather than use Eq. (39) which was based on T6 = 35. and 0 16 Fowler (1960) gives C 12 /N 14 =0 . 01 /~(.4 = 0.05 at a temperature of about 15 million degrees . These values do not change f, ~, or'{ Significantly, but [017 / 0 16]CNO is decreased to O. 072 (a factor of about 2. 5). dashed line in Fig. 31 shows this ratio. The bottom horizontal It is seen that this results in a shift of only about one million degrees in the processing temperature for the destructive case. Thus Fig. 31 shows that for de- structive interference a temperature of about 17 million degree s for the processing temperature in the CNO-cycle appears to be consistent with the present calculations* and that about the same tem2 2 perature would result if the constructive case applied and e e p a were reduced by a factor of 10. A rough estimate of remote-level contributions to S(E) gives an S-value of about 10 kev-barns in the stellar region. This would give a rate of the same order as that calculated for the destructive case and so would not greatly affect the above quoted processing temper ature . In summary we point out that the terrestrial 0 17 / 0 16 ratio can be obtained if *Calculations by Burbidge et al. (1957) would indicate that at temperatures given by the low-temperature intersections of the rate curve with the horizontal dashed lines the mean life of 0 16 is quite long and thus one woulg not expect equilibrium to be reached in the participation of 0 1 in the cycle. No investigation of the CNOcycle under nonequilibrium conditions has been carried out. Equilibrium conditions will almost certainly hold at high temperatures, however, so the present conclusion which excludes processing temperatures greater than 20 million degrees would seem to be a valid one. -63(a) 2 2 4 Constructive interference applies with e e :;:; 10- • p ex This results in a maximum proce.ssing temperature for terrestrial material produced in the CNO-cycle of about 20 million degrees. . . 2 2 -3 (b) Destructive interference apphes wlth e e :;:; 10 p ex gives a processing temperature of about 17 million degrees . Thi s A 2 2 reduction of e e by a factor of 10 here would raise the temperap ex ture to about 20 million degrees. (c) Either constructive or destructive interference applies 2 2 6 and e e :;:; 10- with a processing temperature of about 35 million p ex degrees. It is felt that (c) is rather unlikely and that the best estimate of the processing temperature is about 17 million degrees. It is suggested that nonequilibrium conditions in the CNO-cycle should be investigated in order to determine this temperature more accurately, but it is expected that the temperature will not exceed 20 million degrees in any case. -64APPENDIX A Calculation of Barrier Factors Much of the notation of Lane and Thomas (1958) will be used The partial width r AC for the decay of a compound nucleus here. into channel c via level A may be written 2 2 ;: 2li P 92 = 2P c y,I\.C M a 2 c AC (A.l) c c with (A.2) =k c a c and k2 c Here M E c c = 2Mc E c (A. 3) 1'12 is the reduced mass of the pair of particles in channel c, is their energy of relative motion, a c is the channel radius, '{~c is the reduced width , 9~c is the dimensionless reduced width, and F F c c and G and G c c are the regular and irregular Coulomb functions. are functions of two parameters p Zl Zo e 11 :: c c 11 c 2 c and 11 c where M 1/ 2 (__ c ) 2E c (A.4) Here Zlc and ZOc are the charge numbers of the two particles of the pair c. If we introduce the reduced mass number A, the mass numbers ~ and AO of the pair c, and a characteristic energy Ea defined by -651'12 ::: 20.9 x 10 3 key a 2M a 2 Aa 2 c c c 13 (1 fermi::: 10cm) then we may write E ::: a c (A.5) in fermis (A . 6) 1/2 P (A.7) ::: (E / E ) c c a E The channel radius a c in key (A.8) will be taken to be c (A.9) For the 0 17 (P. a)~4 reaction we consider two channels. the o17 + p channel and the N 14 + a channel. for the energy in the 0 the N 14 + a channel. 17 We let the symbol E stand + P channel; then E + Q is the energy in We let 1 denote the relative orbital angular momentum in units of -Ii for the pair under consideration. We then have (A.10) a::: 5.000 fermi A::: 0.9517 E a ::: 878 key P T) ::: = E l/ 2 29.63 38.87 Ell2 E in key E in key key E in key -66(A.ll) A::: 3.1138 E a = 5.597 fermi _ (E+ Q)1/ 2 = 214 key a P- E,Q in key 14. 63 11 123 - (E+ Q)1/2 E,Qin key r,l _ 58. 52(E + Q)1/ 2 7,l - F21+,al2 Interpolation in the graphs of Sharp, Gove, and Paul (1955) was used to obtain F; + 3%. a;. The graphs can be read to an accuracy of about Tables II and III list the results of the computation of the ratio of the partial width r,l to the reduced width for several energies and ,l-values. e; (the barrier factor) Equations (A.IO) and (A.ll) were used in these computations. In astrophysical calculations it is often necessary to obtain the barrier factors at very low energies where p is very small and 11 is large. This is the case, for example, for the 0 the calculations carried out in Section VI. 17 + P channel in Such a formula has been given, in reference to astrophysical applications, in the paper by Burbidge, Burbidge, Fowler, and Hoyle (1957). The following brief discussion as to how one obtains the formula is based on a comprehensive article by Hull and Breit (1959). For small p one has a;> > F; so that in computing r ,l / Q; we need only consider a,l. One then writes -67x2 and expands G constant . 1 = 8p n = CO. 49 for 7.26 for (A.12) in an asymptotic series in l / n 2 while holding x 2 When all but the fir st two terms in the expansion are dropped and e 2 7r n > > lone obtains (A.13) with + 21{1+1) (21+1) {;)4J where E a (A.14) has been defined in Eq. (A. 5) and K (x) is the modified n Bessel function of the second kind of order n. be valid one should have O!lE« For Eq. (A . 13) to 1. otherwise more terms in the asymptotic series should be kept. We now combine Eq. (A. 6) with these results to find (A.15 ) (1 - O! 1 E) with r 1 having the same units as Ea' In comparing this with Bur- bidge . et al. (1957) one must note that their e: is to be multiplied by 1. 5 to agree with the definition being used here. form for x is A convenient -68x = 0.525 (AZIZOa)1/ 2 a in fermis (A.16) and the quantity b in the exponential in Eq. (A. 15) is given by (A.17) -69APPENDIX B Angular Distribution Factors For the case where only a single level of spin and parity J1T contributes to the observed reaction and where a single orbital angular momentum 1. is effective in forming the compound state and a single orbital angular momentum 1. ' occurs in the decay of the compound state, the general expression for the angular distribution of the reaction (Lane, 1958) reduces considerably. The energy dependence of the cross section factors out and the only remnants of the scattering matrix in the angular distribution function are amplitudes for formation of the states through the various channel spins involved in the reaction (Blatt, 1952). In the 0 nel spins involved in the 0 17 17 14 . (p, a)N reachon there are two chan- + P channel, s = 2 and s = 3; and there is one channel spin involved in the N 14 + a channel, S' =1. We define a quantity r by = probability to form compound state through s = 3 r (B.l) l-r ;; probability to form compound state through s ::; 2 We then write the cross section proportional to an angular distribution function w(9) and find w(9) LL = s {_l)s-l[ (1-r)os2 + ros ] Z{1.J1J, sL) Z(1' J1'J ,lL)PL{cos9) 3 L (B.2) where the P L are Legendre polynomials, 0s2 and 0s3 are Kronecker ° symbols, s I has been set equal to unity, and the Z are defined by Lane and Thomas (1958). They are related to tabulated Z coefficients through -70- _ .1 -l Z-L 1 Z(l J I J ,sL} = 1 Z(1 J I J ,sL} 1 I Z Z l I Z Z (B.3) hi. the calculation performed here the Z coefficients were taken from the tables of Sharp, Kennedy, Sears, and Hoyle (19S4). The distribution factor f3 is given by 41TW(lSZo} f3 =-"----''''---~--- (B.4) 17T Z7Tw(9} sin e de o On making use of the orthogonality of the P and the relation (Lane, L 19S8) (B. S) one finds (B.6) 7T The results for a variety of (J ,J.,l l ) are given below. (0-,3,l): f3 = 1 + (1 ,Z, O): f3 = 1 + (1 ,Z, Z): f3 = 1 + (9r/ 14 - l / Z}P Z (l-,l,l): f3 = 1 - (l/ lO}P Z (1-,3,l): f3 = 1 + (9r/ 10 - Z/ S}P Z + (Z ,O,Z): f3 = 1 + (Z ,Z,Z): f3 = 1 + (ZO/ 39Z}(Sr+ 3}P Z+ (6/ 49}(Sr-4}P 4 (Z-,l,I): f3 =1+ (1/10}(9r - 7}P Z (Z-,l,3): f3 =1+ (4/ 3S)(llr - 7}P Z -71(3+ ,0, Z): {3 + (3,Z,Z): {3 = 1 - (Z/ 49)(17r-6)P Z + (1l/ 49)(5r-3)P 4 (3-,1,3): {3 + (4,Z,4) : {3 = 1 + (1/ 196)(1l3r + 130)P Z - (Z7/ 98)(3r-1)P 4 (4-,1,3): {3 =1 =1 - (Zl/ 140)(9r - 4)P Z =1 + (55 / 84)P Z In the above the Legendre polynomials P Z and P 4 are to be o evaluated for e = 15Z . probability r. r The factors {3 are linear functions of the Thus the limits on {3 are given by setting r = 0 and =1. The se limits, called {3 Z and {3 3' are g iven in Table IV. -72APPENDIX C Energy Spread Introduced by Spectrometer Solid Angle Suppose that a yield curve of counts vs bombarding energy EIB is taken for a case where no reaction occurs except when the incoming particle has energy El = Er and no counts are obtained unless the particle emerging from the target has energy E20=Em-Z2eVt where E m is the spectrometer setting. (We thus have a zero width resonance and a spectrometer with infinite energy resolution.) Refer to Fig. 24 in what follows. Suppose the apparatus is set so that nEls = E r + --=,.;cos 9 (C.l) 1 and nE ZS E 20 =EZ(E r ,9)- cos 9 2 where n is the number of stopping atoms per cm (C.2) 3 and s is the per- pendicular distance from the target surface to the reaction position in the target. If so, then counts are being obtained. the bombarding energy to EIB + AE IB . Now increase The position at which the resonance energy occurs in the target moves to s + As, thus nAsE AE l =-----,.;IB cos 9 (C.3) 1 We assume that in taking the yield curve the magnet setting E being changed linearly with E m is = so that AE 0' AE which guaranerne tees that (C.4) -73where CL is a constant. The energy E I ZO . of a parhcle produced at s + As and emerging at an angle of 9 + A6 is n(s + As)E - cos (6 Expanding and noting that A9 1 _ Z + A9 ) Z Z (C.5) = -A9 Z gives {BEZ) E ZO -EZ(E ,9)+ 0 7 \ r v.. E A9- nEZ(s + As) nE zstan9 z 9 + 9 cos Z cos Z (C.6) A9 1 These particles will be counted if A9 is such that EZO+AE ZO = E~O' Applying this condition to Eqs. (C. Z), (C.3), (C.4), and (C. 6) gives BEZ) nE Z s -+ tan 9 B9 E cos 9 Z Z 1 t (C.7) Note that in deriving Eq . (C . 7) energy variations in the stopping cross sections have been neglected--it being assumed that appropriate energies are chosen at which to evaluate them. Essentially the same relation as Eq. (C. 7) may also be derived by using an expansion and partial derivatives given by Bardin (1961). The maximum energy change allowable that will still produce counts is obtained by setting A9 ;: 59 where 59 is the angular opening of the spectrometer. 59 was calculated from l/ Z 59;: (4(2 /n ) and amounts to 0.074 radians. radians (C.8) The energy spreads AElB calculated from Eq. (C. 7) are g i ven in the column labeled Ag in Table IX . . -74APPENDIX D Energy Straggling In this appendix we derive an expression for the spectrometer yield from a thick target when both energy straggling and spectrometer resolution are taken into account. The case where the bombarding energy E1B is varied over the region of a very narrow resonance is considered. We then discuss the problem of the interpretation of the area under the yield curve. Finally the cal- culated full width at half maximum is compared with what one obtains when a square-root-of-the-sums-of-the-squares law (to be defined) is applied. Let P (E , El;x) dEl be the probability that the incoming parl 1B ticle having a bombarding energy E1B will have an energy between El and El + dEl after penetrating a distance x into the target. Let PZ(EZ,EZS;xl) dE ZS be the probability that the particle pro- duced in the target at position x with energy E Z = EZ(E l ) (deter- mined from kinematics) will have energy* between E dE ZS after passing a distance x, through the target. ZS and E ZS + We shall con- sider the case where the target normal bisects the angle between the incoming proton beam and the spectrometer position; thus x = x'. If we let n be the number of reacting atoms per cm 3 then the number of particles produced per unit solid angle per incident proton in a region dx at x in the target is *The present notation differs from that of Bardin (1961) in that he uses EZO to refer to the general energy of the emerging particles. We use E since our E has already been defined by Eq. (lZ). ZS ZO -75- where IT{E ) is the laboratory reaction cross section per unit solid I angle evaluated at the energy E . I is in the range E Z to E Z The energy of these particles + dE Z at the mom~nt of production. By the time the particles have passed a distance x through the material and have emerged from the target this sharp energy distribution has been spread by the straggling . The number of these particles (per incident particle) having energy in the range E to E ZS ZS + dE ZS is An integration over EI gives the observed particle spectrum from a target lamina dx at x. A second integration, this time over x, gives the complete observed particle spectrum from the entire target. 6 E m Finally, an integration of E ZS over the energy acceptance = ZEm / R of the spectrometer (we assume a rectangular dis- tribution for the spectrometer energy acceptance) gives the laboratory yield Y L{E IB spectrometer and R lution. Here Ern is the energy setting of the = p / Ap is the spectrometer momentum reso- We thus have EIB YL{E IB ) ). =n EZO+ Em / R X S S S dx dEl o 0 dE ZS PI{E IB , EI;x)PZ{E Z ' EZS;x)o-{E I ) EZO-Em / R {D. I) The normalization condition on the probabilities P{E, w;x) is E S P{E, w;x) dw o =I (D . Z) -76For the case of a very narrow resonance in the cross section we write (D.3) where the area "under" CJ r the o-function is written 7rCJ r / 2 sin ce if r is the cross section at the resonance and r is the total width this is just the area given by the single level formula fo r a narrow level. What is meant here by a very na rrow resonance is that r- 0 and CJr - 00 in such a way that CJ r is constant. r (D. 3) is the resonance energy. performed. E r in Eq. The El integral can then easily be We prefer here not to use x as a variable but to mea- sure depth into the target by means of the energy loss S to the incoming particle s. It is a good approximation to neglect the energy variation of the stopping cross section. We then have where the stopping cross section has been evaluated at the resonance energy and n 3 s is the number of stopping atoms per cm . Equation (D.5) where ~ is the target thickness in energy units and E2r Equation (D.5) assumes E 1B = E 2 (E r ) . > Er' otherwise the yield is zero. The next step is to as sume that the straggling probabilities are given by Gaussian distributions with means given by E1B - Sand -77E2r - E2r g/ E and with rms deviations t:. given by lr (D. 6) with A 4 __ 4_7r_e_ ' \ r Elr L Z. (D.7) 1 i In the above 'expression for A , E is the stopping cross section r lr per molecule evaluated at E atoms (L Zi ::: 36 for NiO). r , and Z . is the charge of the stopping 1 This expression for the standard devi- ation was originally given by Bohr (1915). The condition for validity of the Gaussian approximation has been given by Rossi (1952) and is that the rms deviation t:. must be large compared with the maximum transferable energy in a single collision Q max yet small compared to both the average energy at distance x and the energy loss. Q .max is given by (D.8) where M and E are the mass and energy of the particle under consideration. In the present experiment it is found that the above criterion is fairly well satisfied for the alpha particles but only poorly satisfied for the protons. This is reflected in the disc rep- ancy between the calculated and experimental yield curve at the high-energy end (Figs. 12 to 18). The Gaussian assumption yields (D.9) -78and (D.10) On combining Eqs. (D.5), (D.9), and (D.10) one notices that the E integral can be performed leading to error functions (erf). ZS This gives (D.11) with -(E -E +£) [ r lB Z (D.1Z) where (D.l3) The error function is defined by =-Z S .fr. 0 x erf(x) e _t Z (D.14) dt The Burroughs ZZO computer was programmed to cb the integral I(E 1B ) [Eq. (D.1Z)] for the seven narrow resonances. trometer setting E was changed. m was changed as the analyzer The spec- energy E e The relation between these quantities was very closely linear so we write [see also Eq. (C.4)] E m =0£ e +[3 (D. 15) The parameters used for each resonance are listed in Table VIII. -79The resonance energies were chosen so that the peak of the calculated integral and the peak of the experiInental points fell at the same value of E lB . These resonance energies are equal to those calculated from Eq. (13) to within a few tenths of a kev. The in- tegrand in Eq. (D.lZ) was found to fall to very small values before ~ reached the actual target thicknes s. = 20 kev was A value:=: found sufficient and was used in all the calculations . After the integral had been calculated as a function of ElB use was made of Eq. (D.ll) to obtain the yield. A value of CT r r was chosen that normalized the calculated yield to the experimental yield at the maximum value. These calculated results are shown as dashed curves in Figs. lZ to 18. We next discuss the problem of the relationship of the area Ay under the yield curve to the level parameter s. Ay is defined by (D .16) A somewhat different area A'y is obtained if one writes the yield as a function of the reaction energy El instead of the bombarding energy. We have (D.17) The relationship between the two areas can be found from Eq. (B), and is € Z/€lr+aEz/aEl € Z/€lr + ct. A'Y (D . 18) -80where a is defined in Eq. (D.1S). The area under the calculated yield curve was also computed on the Burroughs 220. The quantity A y / (rrrr1r/ 2) resulting from this computation is given in the second column of Table X. We now obtain the corresponding relations for the case of no straggling, using lower case symbols for this case. rr(E) = We let r 2/ 4 rr r l (E -E )l + r / 4 (D.19) r If we then substitute Eq. (D. 19) and Ii -functions for the straggling probabilities into Eq. (D.l) one obtains nrrrr -1 n tan lr s (D.20) 2/': This result has also been given by Fowler et al. (1948). The quan- tity £1 is given in Eq. (15) and is a function of El through Eqs. (D.1S) and (13). We define the areas a ay y and a' by Y S = YL(E1B ) dE1B (D. 21) and (D.22) The relation (D.23) also holds here. If one then writes £1 explicitly as a function of El the integral a' can be performed giving y -81(D.24) with (D. 25) G is quite small in the present experiment so the correction term in Eq. (D. 24) which involves G will be neglected from now on. Ap- plication of Eq. (D. 23) then gives a 1[(1" r 2nE (max) m = r /2 n s E1r R (E 2 r 7E1r + a) y (D.26) The results of applying Eq. (D. 26) to the resonances are also listed in Table X. The fact that the numbers in the two columns are so nearly equal shows that as far as the yield area is concerned straggling can be neglected. This fact is used in the disc ussi on in Sec- tion V -C4. Brief mention should also be made of the fact that if one does not restrict the E 2S integration in Eq. (D.l), but requires that one observe the entire outgoing particle spectrum, then the E gives unity by virtue of Eq. (D.2). 2S integral The situation then reduces to that considered by Gove (1959) in which he shows that the area is independent of energy spread in the beam or straggling of the incorning particle s. As a final topic in this appendix we briefly investigate the prob lem of compounding experimental widths from several sources into a total width. A rule for compounding experimental widths that • -82- has been used in the past (Cohen, 1949) is to approximate the various distribution functions with Gaussians having the means and rms deviations of the true distributions. The result of folding these Gaussians together then yields a Gaussian with an rms deviation equal to the square root of the sums of the squares of the rms deviations of the individual distributions. This type of rule can be checked against the present exact calculation. We let Ll.Ep and Ll.EO' stand for the straggling widths of the protons and alpha particles respectively. We use Eq. (D.6) evaluated at the value of £ which gives the maximum yield. We thus take Ll.E 2 2 = = (D.27) Ll.E / 4 A [E1B{max}-E ] par r Note that Ll.Ep and Ll.EO' are not necessarily the direct contributions to the spread in Y L (E 1B ) from straggling. The direct contributions, denoted by IT P and ITO' must be calculated, and depend in part on the rate at which the spectrometer setting is varied with respect to bombarding energy. We first note that the contribution Ll. m to the width of Y L{E 1B ) from the spectrometer window may be found from £1' as given in Eq. (15), and from Eq. (13). trometer window. LI. The values of LI. m £1 is just the width of Y L{E ) due to the specl We thus have (D.28) m are listed in Table IX. The spectrometer resolution function is assumed to be rectangular so that IT m 2 =LI. m2 / 12 (D.29) -83where CT m is the rms deviation contribution to the width of the yield curve from the spectrometer energy window. We next compute CT by as suming that proton straggling is the p only source of width. Monoenergetic protons are i ncident on the tar- get surface and pass into the target. tance x r Let the mean energy at a dis- in the target be the resonance energy E the protons have an energy spread Ll.E p r . At this distance due to straggling; however, we consider the cross section to have zero width, and therefore only protons with energy E r will produce alpha particles. distance in the target, say at x E r , but due to the spread Ll.E energy E r p r + dx, the mean proton energy is not there are protons at x , and which will thus initiate a reaction. therefore, that along a length Ll.x at x energy E r At a slightly different r r + dx which have It can be seen, there are present protons of , which can produce alpha particles of energy E length LI.x is related to the width Ll.E Ll.x The energy E ZS p Zr = Ll.E p / nE 1r (D. 30) of an alpha particle emerging from the target and hav- nx E Zr' and the energy spread Ll.E ZS The through ing been produced at a distance x in the target is given by E by Ll.E . ZS ZS =EZr - of these alpha particles is given = n E ZrLl.x. On using Eq. (D. 30) one then finds (D. 31) which gives the spread in alpha particle energy due to proton straggling at a bombarding energy E 1B . Suppose, then, that the spectrometer is set to detect the alpha-particle energy at the edge of this spread , i. e. suppose we have -84E 20 :; E2r - (ElB-E r ) E 2 /E lr - .6E 2S / 2 Then let us increase ElB by .6E lB and E 20 by .6E 20 :; .6E (D.32) m :;a.6.E lB so that we have (D. 33) Equations (D. 32) and (D. 33) guarantee that the change .6E bombarding energy just covers the spread .6E spectrum. 2S lB in the proton in the alpha particle On subtracting the two equations and using Eq . (D. 31) one obtains :; .6E p E 2r/ E 1r (f p E 2/Elr + We have set (fp :; .6E lB (D. 34) a The quantities (f 2 a r e given p • by definition of (fp in Table IX. A similar calculation yields (fa. Here we assume that the only wid th comes f r om the alpha-pa r ticle straggling. The alpha particles pro- duced in the target with energy E2r emerge with a spread .6E (D. 32) and (D . 33) then hold with .6E .6E lB 2S replaced by .6E a. a. Equations If we set :; (fa and use Eq . (D . 27) we find 2.6E (fa:;E2/E~r+a The quantities (f; (D.35) are listed in Table IX. We define .6 2 2 1/ 2 :; 2. 35 [(f2 + (f + (f ] sum p m a (D.36) and compare this full width at half maximum with the true value .6 . t The factor 2.35 is the ratio of the full width at half maximum to the rms deviation for a Gaussian. The values of .6 sum are given in Table IX along with the values of .6 • which were determined from Figs. 12 to 18. t agreement is quite good. The -85REFERENCES K. Ahnlund, 1957, Phys. Rev. 106, 124. F. Ajzenberg-Selove andT. Lauritsen, 1959, Nuclear Physics 11, 1. S. K. Allison, 1958, Revs. Modern Phys. ~, 1137. E. Almqvist, D. A. Bromley, and J. A . Kuehner, 1958, Bull. Am. Phys. Soc. 3, 27. R. K. Bardin, 1961, Ph.D. Thesis, California Institute of Technology . S. Bashkin, R. R. Carlson, R. A. Douglas, and J. A. Jacobs, 1958, Phys. Rev. 109, 434. J. M. Blatt and L. C. Biedenharn, 1952, Revs. Modern Phys. 24, 258. N. Bohr, 1915, Phil. Mag. 30, 581. R. O. Bondelid and C. A. Kennedy, 1959, Phys. Rev. 115, 1601. A. B. Brown, C . W . Snyder, W. A. Fowler, and C. C. Lauritsen, 1951, Phys. Rev. 82, 159. E. M. Burbidge, G. R. Burbidge, W. A. Fowler, and F. Hoyle, 1957, Revs. Modern Phys. ~, 547. R. R. Carlson, C. C. Kim, J. A. Jacobs, and A. C. L. Barnard, 1961, Phys. Rev. 122, 607. E. R. Cohen, 1949, Ph.D. Thesis, California Institute of Technology. G. A. Dissanaike, 1953, Phil. Mag. 7.., 44, 1051. J . P . Elliott and B. H. Flowers, 1955, Proc. Roy. Soc. (London) A229, 536. J. P. Elliott, 1958a, Proc. Roy. Soc. (London) A245, 128. J. P. Elliott, 1958b, Proc. Roy. Soc. (London) A245, 562. F. Everling, L. A. Konig, J. E. Mattauch, and A. H. Wapstra, 1960, Nuclear Physics~, 529. W. A . Fowler, C. C. Lauritsen, and T. Lauritsen, 1948 , Revs. Modern Phys. 20, 236. W. A . Fowler, 1960, Memoire Soc. Roy. Sciences de Liege~, 207 . -86W. A. Fowler, J. L. Greenstein, and F. Hoyle, 1961 (to be published). This article is referred to by the abbreviation FGH. H . E. Gove, 1959, "Resonance Reactions, Experimental, " in P. M. Endt and M. Demeur, Editors, Nuclear Reactions (Interscience Inc., New York), Chap. VI, p. 259. L. H. Gray, 1943, Proc. Cambridge Phil. Soc. 40, 72. D. F. Herring, R. Chiba, B. R. Gasten , and H. T. Richards, 1958a. Phys. Rev . 112. 1210. D. F.Herring. 1958b. Phys. Rev. 112. 1217. N . P. Heydenburg and G. M. Temmer. 1953. Phys. Rev. ~. 89. S. Hinds and R. Middleton. 1959. Proc. Phys. Soc . (London)~, 721. M. H. Hull. Jr. and G. B reit. 1959. " Coulomb Wave Functions," Handbuch der Physik (Springer-Verlag, Berlin). vol. XLI/ 1, p. 408. E. Kashy, P. D. Miller. and J. R. Ris ser. 1958. Phys. Rev. 112. 547. A. M. Lane and R. G. Thomas. 1958. Revs. Modern Phys. 30 , 257 . J . B. Marion. 1961. Revs. Modern Phys. ~. 139 . E.A. Milne. 1953. Ph.D. Thesis . California Institute of Technology . F. S. Mozer, 1956. Ph. D. Thesis. California Institute of Technology. A. O. Nier. 1950. Phys . Rev. 7.2. 789. W. R. Phillips. 1958. Phys . Rev . 110. H08. R. L. Platzman , 1952. " Influences of Details of Electronic Binding on Penetration Phenomena. and the Penetration of Energetic Charged Particles through Liquid Water. " in J. J. Nickson, Editor. Symposium on Radiobiology (John Wiley and Sons . Inc .• New York). Chap. 9. p. 139. P. C. Price. 1955. Proc. Phys. Soc. (London) A68. 553. M. G. Redlich. 1954. Phys. Rev. 95. 448. M. G. Redlich . 1958. Phys. Rev. 110. 468. H. K. Reynolds. D. N . F. Dunbar. W. A. Wenzel. and W . Whaling. 1953 . Phys. Re v. 92. 742. B. Rossi, 1952, High-Energy Particles (Prentice-Hall , Inc. , New York) . -87W. T. Sharp, J. M. Kennedy, B. J. Sears, and M. G. Hoyle, 1954, Atomic Energy of Canada Ltd., AECL-97. W. T. Sharp, H. E. Gove, and E. B. Paul, 1955, Atomic Energy of Canada Ltd . , AECL-268, 2nd edition. E.A. Silverstein, G. Hardie, L. Oppliger, and S. Salisburg, 1960, Bull. Am. Phys. Soc. ~, 405. C. W. Snyder, S. Rubin , W. A. Fowler, and C. C. Lauritsen, 1950, Rev. Sci. Instr. 21, 852. W. Whaling, 1958, "The Energy Loss of Charged Particles in Matter," Handbuch der Physik (Springer-Verlag, Berlin). Vol. XXXIV, p. 193. -88aTable 1. Contributions to error in cross section. The quantities needed to obtain the absolute cross section are given along with their estinlated rrns deviations in percent. These quantities appear in Eqs. (21.) and (22). cussion of these errors see text p. 31. ff. For a dis- -88Table I Contributi ons to Error in Cross Section Quantity Error (percent) F 200 17 0 concentration 2.6 1.0 Q 0.4 ai R 3.6 kinematic s 0.0 E 0.2 m in NiO) 3.0 E (a in NiO) 10.0 E (p E18E/8E l + E 2 9.0 cross section 10.0 -89aTable II. B arrier factors for the 0 barrier factors values. + P channel. The r;1e; are given in kev for various i.- The proton energy E the c. m. energy E in the 0 in kev. 17 17 p in the laboratory and + P channel are also given The calculation of these factors is described in Appendix A. See text pp. 35, 38 , 46, 55 , 66. -89Table II Barrier Factors, 0 E p E 1.;:; 0 1. ;:; 1 17 +p 1.;:; 2 1. ;:; 3 1. ;:; 4 1350.6 1275.0 407 1274.0 1202 . 7 924 333 31. 1 1. 70 0.0457 1246.7 1176.9 865 308 27.5 1. 54 0.0395 1101. 3 1039.7 622 195 15.9 0.843 0.0176 927.1 875.2 361 101 7.15 0.294 0.00538 825.0 778.8 241 60.2 4.06 0.153 0.00258 747.2 705.4 173 37.4 2.56 0 . 0841 0.00121 672.4 634.8 110 23.3 1. 42 0.0406 5. 6 8 x 10 -4 518.5 489.5 37.2 5.82 0.377 0.00675 8.35x10 -5 -90aTable III. Barrier factors for the N barrier factors values. r;/a; 14 + a channel. The are given in kev for various L- The energies listed are those for the 0 channel, E p 17 +P being the proton energy in the laboratory and E being the c.m. energy. are given in kev. Both of these energies The c . m. energy in the N channel is given by E + Q. +a The calculation of these factors is described in Appendix A. 38, 53, 55, 66. 14 See text pp. 35, -90Table III B arrler . F actors, N 14 + a E E E 1. =0 1. =1 1. =2 1. =3 1. =4 1350.6 1275.0 1274.0 1202.7 372 242 102 28.6 4.90 1246.7 1176 . 9 354 230 99.3 25.9 4.52 1101.3 1039.7 271 177 67.4 17.1 2 . 82 927.1 875.2 185 114 40.8 9.47 1. 36 825.0 778.8 140 85.2 29.4 6.54 0.887 747.2 705.4 115 68.0 23.6 5.00 0 . 644 672.4 634.8 86.4 47.7 16.6 3.34 0.400 518 . 5 489.5 49.0 25.3 8.52 1. 55 0.185 280 92.0 2.32 40.1 1. 58 -10.7 1. 02 -9laAngular distribution factors 13 , 13 for i so3 2 Table IV. lated levels in the reaction Ol7(p,a)~4. J 1T refers to the total angular momentum and parity of the state in F18 The orbital angular momentum in the 0 denoted by L and that in the N by L I . 14 17 + P channel is + a channel is denoted The quantitie s 13 2 and 13 3 are for formation of the compound state through channel spins 2 and 3 respectively. The actual 13 for the level must lie between these two values. In the cases where only one channel spin is possible, 13 is unique and is given in the appropriate column. The calculation of these factors is described in Appendix B . See text pp . 36 , 38 , 71. -91Table N Angular Distribution Factors J11" l I' 0+ 2 forbidden 0 3 1 1+ 2 0 1. 00 1. 00 1+ 2 2 0.66 1.10 1 1 1 0.93 1 3 1 0.73 2+ 0 2 1. 00 2+ 2 2 0.84 0.74 2 1 1 0.54 1.13 2- 1 3 0.46 1. 31 3+ 0 2 3+ 2 2 1. 09 0.75 3- 1 3 1. 40 0.50 4+ 2 4 0.53 1. 00 4- 1 3 13 2 13 3 1. 00 1. 34 1. 00 1. 45 -92aTable V. Barrier factor coefficients. The coefficients a , b , and c have >been used to fit polynomials in E n n n to the barrier factors in two energy regions. pp. 38, 53. See text -92Table V Barrier Factor Coefficients n a 0 -4.974 x 10 1 1. 902 x 10 2 -2.723 x 10 3 1. 735 x 10 4 -4.001 x 10- 9 • b n 3 1 -2 -5 c n -4.586 x 10 1. 564 x 10 -1. 957 x 10 1. 088 x 10 3 1 -2 -5 -2.197 x 10- 9 n 1.126 1. 011 x 10 3.093x10 -2 -5 -93aTable VI. Partial widths for the two broad resonances occurring at c. m. energies of 1035 and 1203 kev (lab energies of 1096 and 1274 kev) as determined by the fitting of a simple sum of two Breit- Wigner expressions to the experimental data. E r The c. m. resonance energies and the partial widths rand r p a are given in kev. The widths in Group I were those used in obtaining the dashed curve shown in Fig . 22. Group II or a combina- tion of Ia with IIb or Ib with IIa could also have been used without destroying the fit. 55. See text pp. 39. 47. -93Table VI Partial Widths 2 p 9 a rp ra 1035 0.013 0.470 2.5 82.5 Ib 1203 0.194 0.060 64.5 14.5 II 1035 0.430 0.015 82.4 2.6 1203 0.044 0.267 14.5 64 . 5 Group E I a a r 9 2 -94aTable VII. SUIIlmary of level parameters as determined in the present experiment. The quantity j3gr r I ris a p given in ev and all other quantities are expressed in kev. See text pp. 39, 43, 45, 47, 55. -94Table VII Swnroary of Level Parameters ra r E (Lab) r E (c.m.) r E ex (3g~ 518 + 2 489 + 2 6086 50 + 5 <2.0 672 + 2 635 + 2 6232 43 + 4 <2.0 747 + 3 705 + 3 6302 100 + 10 825 + 3 779 + 3 6376 23 + 2 <4.5 927 + 3 875 + 3 6472 39 + 4 <1. 2 1096 + 6 1035 + 6 6632 1101 + 4 1040 + 4 6637 36 + 4 <3.0 1247 + 5 1177+5 6774 150 + 16 10 + 3 1274 + 5 1203 + 5 6800 1335 + 10 1260 + 10 6857 r(c.m.) 3.1+1.4 85 + 5 79 + 5 -95aTable VIII. Straggling-integral parameters needed in the calculation described in Appendix D. Ar' and j3 are expressed in kev. E r , E Zr ' Elr is given in Z and Elr / E Zr and a are dimensionless. 10 -5 ev-cm, See text pp. 41, 78 . .. -95Table VIII Straggling-Integral Parameters Elr E 2/Elr A r a f3 518.5 26.03 4.950 0.3613 0.6429 783.9 1155.7 672.4 22.51 5.664 0.4176 0.6870 731. 2 1239.9 747.2 21. 20 5.981 0.4434 0.7137 676.0 1281. 3 825.0 19.99 6.298 0.4703 0.6101 761. 9 1324.5 927.1 18.72 6.661 0.5022 0.5070 880.3 1381. 5 1101. 3 16.92 7.228 0.5556 0.5234 819.1 1479.3 1246.7 15.70 7.643 0.5987 0.5061 869.9 1561. 3 E r (Lab) E 2r -96aTable IX. Width contributions to the yield curves given in Figs. 12 to 18. (T~ and (T~ are given in kev 2 and the remaining quantities are expressed in kev. The calculation of several of these width contributions is described in Appendix D. 73, 83 ff. See text pp. 40, • -96Table IX Width Contributions 2 ITa II stun llt 1. 90 0.31 4.3 4.5 3.49 2.39 0.31 4.5 4.6 3.47 3.38 3.02 0.35 4.9 4.9 1.0 3.41 3.38 3.32 0.33 5.0 5.1 927 1.1 3.48 3.51 1. 78 0.16 4.0 4. 1 1101 1.0 3.29 3.31 5.17 0.22 5.9 5.9 1247 1.1 3.37 3.39 3.79 0.25 5.3 5.2 2 E (Lab) r lle ~l II m -L 518 0.95 3.79 3.72 672 0.98 3.57 747 0.95 825 IT -97aTable X. Calculated area of narrow resonances . resonance energy E r is given in kev. The The other two quantities are given in crn -2 and are proportional to the area under a yield curve for a narrow resonance. Both the case of straggling of the incident and outgoing particle and of no straggling are given. The calcula- tion of these quantities is described in Appendix D. See text p. 80L -97Table X Calculated Area of Narrow Resonances E r (Lab) a Ay 1f (l' r j 2 r (straggling) y rj2 r (no str aggling) 1f (J" 16 518.5 0.5695 x 10 672.4 0.6205 0.6201 747.2 0.6387 0.6391 825.0 0.6809 0.6808 927.1 0 . 7470 0.7458 1101. 3 0.7890 0 . 7905 1246.7 0 . 8705 0 . 8701 0.5699 x 10 16 -98aTable XI. E ex Reported c. m . widths of the Fl8 levels at = 6632 and 6800 kev. Previous determinations of these widths are compared with those of the present work. Widths are given in kev. See text p. 45. -98Table XI Reported C. M. Widths of the F18 Levels at E Source (Heydenburg, 1953) ex = 6632 and 6800 key 6632-kev Width 6800-kev Width 27 + 4 93 + 8 (Ahnlund, 1957) 90 (Herring, 1958b) 93 + 5 101 + 5 (Kashy, 1958) 59 + 8 74 + 8 Present work 85 + 5 79 + 5 Table XII. Stellar reaction rates. Various reaction rates as a function of temperature i n millions of degrees are given . The calculations are described in Section VI. See text p. 60f. I -D -D Pl I Table XII Stellar Reaction Rates p T6 log P17 c 1 17 HO (cons. ) log (deat. ) P 17 1 17 log 1 17 HO HO (cons.) log P 16 H I 01{) (dest.) log o 17 P 16 16 o p 17 o log o 17 16 P16 p 17 (cons. ) (de st. ) 5.0 -54 . 770 -55.397 -87.981 -54.770 -55 . 397 -56.901 -2.130 -1. 504 10.0 -46.387 -47.524 -55.659 -46.387 -47 . 524 -48 . 355 -1.968 -0.831 12.5 -44.051 -45.521 -49.249 -44.051 -45.521 -46.007 -1. 955 -0.486 15.0 -42.272 -43.828 -44 . 998 -42.271 -43.797 -44.219 -1. 947 -0.421 I -.J:) -.J:) I 20.0 -39 . 555 -40 . 514 -39.723 -39.330 -39.658 -41 . 613 -2.282 -1. 954 25.0 -37.550 -38.222 -36.591 -36.546 -36 . 574 -39.763 -3 . 216 -3.188 30.0 -34.525 -38.353 -3.828 -3.828 35.0 -33.065 -37.229 - 4.164 -4.164 40.0 -31. 981 -36.302 -4.321 -4 . 321 45.0 -31. 148 -35.521 -4.373 -4.373 F ·igure 1. Proton counts vs fluxmeter setting V elastic scattering of protons by Ni. for the The figure compares a clean Ni target with an oxided Ni target. bombarding energy E1B was 1. 005 Mev. 10£. m The proton See text pp. 8, .....I o o PI I Figure 2. a vs fl. as determined from data in Fig. 1. a and fl. are defined in Eqs. (3) and (4). See text p. lOf. I ..... o..... III I -101- N (!) o u.. 0'1 CD I'- o o o , ~igure on NiO. 3. Alpha-particle pulse height spectrum for protons The solid histogram is that obtained with a target enriched in 0 17 and the dashed histogram is that obtained with a target made with natural oxygen. E1B = 1. 342 Mev, V m For this spectrum = 498.6 mv, and Q = 276 fJ.coul. See I ..... o N P> textp.17. I -102- , , , I ' , , , I ' , , , I ' , , , I '.' , , I ' I ..- I , "T" 0 I - Z 0 1 L(') 0 - -... 0 '"j II ~ a.. II) !Xl 1 <t I -I 0 1 en CD I r 1 I (X) .... I I I'-- r' ..J I !.OW I I Z Z IO~ I I u ~ I 1 r' I ..., .J C\I I ,....~ I I L. I I I I 10 C\I I I I I I o o I I I I I I I I I I o 10 S.lNnO~ I I I I I 10 C\I I I I ,." I I , v I Figure 4. Uncorrected data taken at the 672-kev resonance showing counts per integration vs analyzer potentiometer setting V . e Data from both the enriched target and natural target are shown. These data were used to obtain the yield curve of Fig. 13. See text pp. 18, 44. I ...... o VJ II> I I- I -.oJ ::> o I- ::i.. I- ~ t- u I T I • ENRICHED NICKEL OXIDE TARGET · 017 (p,a) N o NATURAL NICKEL OXIDE eLAB =150 TARGET I - I4 0 - C\I - ll.. 200 I- o Z I- r I- o <l 0::: J • I- (!) r W r I- Z 0::: W 0.. 1001- Z ::> o 0 1 ~ I I I •• t- • • C/) r I : •• • • • II- u .. • I- 0 • • I 0 • • • I ~ ::r 0 o 0, o FIG.4 .670 ELECTROSTATIC . ,• 0 • o o• - __1 -L .660 ,I .680 .690 ANALYZER POTENTIOMETER SETTING (DECI V.) " Figure 5. Iron oxide target profile showing alpha-particle counts vs fluxmeter setting V m . Data from both the target back and the oxided target front are shown. were used to obtain ,Fig. 19. These data For this profile E1B Mev and Q = 88.6 fLcoul for each point. =1. 280 See text pp. 19, I 22, 30. ...... o ~ '" I • • TARGET FRONT 0 17 (Pta) N 14 6 TARGET BACK BLAS=150 800.- 0 (f) ~ III z ::::> o 600 I u w ....J u - II ~* ~ a: <t 400 a.. I I <t f :I: a.. 1 ....J <t 200~ I ! 1 ! I I~ 520 FIG.5 • • ~I 515 I I I 510 505 500 FLUXMETER SETTING (mv) 495 Figure 6. Iron oxide target profile showing alpha- particle counts vs fluxmeter setting V m . The circles, triangles, and square are data taken with different reference profiles. The e rror bars have been omitted from the data obtained when the back of the target was bomb arded. tain Fig. 20. Q The data in this figure were used to obFor this profile ElB = 854 kev and = 2850 f.Lcoul for each point. See text pp. 20, 22, 31. I I-' o I.1t III I -105- It) 0 .z 10 , tI II It) 0 - 0 - « a: ,.. - V CD •• ...J OQ) • - • • 1 • ~ 0 It) It) > It) It) It) • • • ~ I--O----i (!) -z ~ ~ • w en <l 0 • • E (0 It) 0 w ~ W ~ • • a:: x ::::> ....J u. It) t-<J--t • • • <l (0 It) o,... It) . <.!) o It) o o S.lNnO~ 3l~I.l~:n1d-'i1Hdl'i1 u. ;: Figure 7. Iron oxide target profile showing alpha- particle counts vs fluxmeter setting V m . Data from the target back and the oxided target front are shown. For this profile E1B :: 1. 005 Mev and Q :: 713 f.Lcoul for each point. See text p. 20f. I ..... o r;r. P> 200 • TARGET FRONT o17 (p,a)N 14 o TARGET BACK eLAB =150 0 C/) I- Z ~ o u W -1 U I0:: I .~! 100 i <J. 1 <t :::L a.. -1 <t !• 50 • f • 560 FIG.? • 550 •• 0 540 FLUXMETER 0 • • • • ! • 0 530 520 SETTING (mv) •o 570 I Figure 8. Alpha-particle pulse height spectrum for protons on the iron oxide target. The solid histogram is for the target front and the dashed histogram is for the target back. V m For this spectrum EIB = 955 kev, =544.0 mv, and Q = 1430 I-Lcoul for each point. See text p. 22. .....I o --J III I · 107 I I I I •-Z I I I I I I I I I I 1 T I r I I I I 0 -.. - 0 10 0 - II a.. ,.. -0 o m ct ....J Q) r 1 ! 1 r--------- . . . . 1 1 L _____ , I 1 , --., ...1 ..... I 1 ! I I 1 I I I o v I I I I I I I I I o I o rt) SlNno~ '" I I I I I o I I I I Ir 1 1 · Figure 9. Alpha-particle pulse height spectrum for protons on the iron oxide target. The solid histogram is for the target front and the dashed histogram is for the target back. V m For this spectrum E1B =854 kev, =560.0 mv, and Q = 2850 jJ.coul for each point. I ..... o CX) See text p. 22. P> I -108I I I •-Z I I I I I I I I I I T T I T I I T I 0 - 0 II ~ .... en 1 1 0 \() 0... I o m <I ...J IT" 1 Ct> 0 J r I 1 , I I 1 1 I .-J oDW -----, Z Z I 1O<t I U I I 1 1 , I-- 1 1 _.J I' I I LL..... , 1 ~ I I I o V I I I I I I o If') I I I I o (\J I I I I I o I I I I F~gure 10. Alpha-particle charge equilibrium fraction in solids. F 200' F Ioo ' and F 000 are the equilibrium fractions of He++, He+, and He respectively. are from the review by Allison (1958). Data See text p. 26. I o'""" -.0 1\1 I -109- 0 0 10- -> W ->~ (!) a: C\I W LL.. Z W OW 0...1 0(,) --.... a: <t a.. I <t ~ a.. ...1 <t 0 0 10 10 o NOll.~'j~.:I v-Jnl~8Illn03 ,. Figure 11. Laboratory differential cross section (TL(9 ) L vs proton energy E . 1 The experimental resolution was about 5 kev due to spectrometer resolution and straggling, and the data in this figure have not been corrected for this resolution. shown. Data from both types of targets are The ordinate scale is accurate to 100/0. See .... .... o I P> I text pp. 24, 29, 31, 36, 44, 47. lOt- -z m<t <to ..J<t a:: Zw or rCl> u' WCl> Cl>Z a:: CI><t Cl>m a:: ....J u..J • o. o • • 5~ ••• .- 0 f::. • 0 • • • .0 5 t ~ 0 0 • 1250 1300 - 1350 • 1400 • 1450 850 1500 • 0 900 1000 950 1050 1100 • .- o· ~ • \.1 FIG.II • • •• 0 0 • 500 0 o. :. 450 II • 0 - j~ 1550 0 0 ~. 5i- • . . · · ..... -...- ./:f'... . • •• • • • ..•• •• ...- • c! o· 0 L IRON OXIDE TARGET 8 LAB =150 0 t I4 • NICKEL OXIDE TARGET • ~o.- ~ ° • 017(p,a) N I I I 550 I I I I I I I I I cJ 600 650 700 • 750 PROTON REACTION ENERGY, LAB (kev) t::; Figure 12. Laboratory alpha-particle yield Y L vs proton bombarding energy ElB near the 5l8-kev resonance. The error bars indicate statistical errors only. The dashed curve has been calculated from the theory given in Appendix D. 39, 41£., 47, 77, 79, 84. See text pp. 29, 31, I ..... ::: III I Z <t I4 0 OI7(p,O) N a: 4 w 8 LAS =150° <t ~/rI,\ I- , (f) Z 0 I0 I a: 3 a.. " <t I a.. -'0 fl 1 I \ I,' (f) ...J _<t \ I I ~ I 2 I - \ I 0 ...J 1; ->- I W \ II I ...J <t \ " CD <t ...J \ \\, I I / a.. \\1 \ / <t \1 - 520 FIG . 12 525 530 PROTON BOMBARDING ENERGY, LAB (kev) 535 Figure 13. Laboratory alpha-particle yield Y L vs proton bombarding energy E1B near the 672-kev resonance. error bars indicate statistical errors only. The The dashed curve has been calculated from the theory given in Appendix D. See text pp. 29. 31. 39.41£ .• 47. 77. 79. 84. I ...... ...... tv III I -I I 2- > Q) .. -.. a.. ...- « .lI: 0 Z ( II 0 <X 10 CS ....J II t--+-f III / / ...J 0 tr CD W ..- Z -- W (!) z J... ...- ...- a .... ,......1 • 1 • 1 1 ,....... . >(!) 1 ....... £r <X (II -- -- • -- ~ o 1 --I • --- (II -~ -- - -- (N'tl a't~31S -N010~ d /S 't Hdl 't 11_0 I) al31A 't H dl't 8'tl z o ..... o tr a.. Figure 14. Laboratory alpha-particle yield Y L vs proton bombarding energy E1 B near the 747 -kev resonance. The error bars indicate statistical errors only . The dashed curve has been calcu- lated from the theory given in Appendix D. text pp. 29, 31, 39, 4lf., 47, 77 , 79, 84. See I '-' '-' v.> II> I Z <t o <t Q: w 0 17 (Pta) N 14 5 eLAB = 150 l(/) I Z o 4 o I- I I Q: a.. "" <t I' J: 3 a.. I -.l <t I '0 I I -o I~\ 1 I (/) 2 <t m <t 1 \ \ I -.l 01 \ I a.. \ \ I <t \ \ \ I J: \ \1 I >- \ \ II -.l W ! \ II \ / -.l 750 FIG. 14 0 755 760 PROTON BOMBARDING ENERG~LAB ~ Figure 15. Laboratory alpha-particle yield Y L vs proton bombarding energy E1B near the 825-kev r e sonance. error s bars indicate statistical errors only. The The dashed curve has been calculated from the theory g iven in Appendix D. See text pp. 29, 31, 39, 4lf., 47, 77, 79, 84. I ..... ..... 0I>- l» I -I 1 4- > Q) •Z 0 0 II -.. - ,... n. 0 .:tt. 0 / It) • m <{ • ...J <t> • ,....../ , / •/ ./ - I,,' 1/ 1/ ./ ./ ......- " .....- .. >(!) a:: w 1 Z /' W (!) Z Cl ( " II) « ...J a:: « II) 1 ""'- ...... • ~ - - -, ..... • • - ~ 0 II) ...... ...... z 0 r- ........ 0 a:: n. • o 10 o (N'tIO't~3.LS-NO.LO~d/S'tHdltfll_OI) Ol31A 'tHdltf 8tfl Figure 16. Laboratory alpha-particle yield Y L vs proton bombarding energy E1B near the 927 -kev resonanc e . e rror bars indicate statistical errors only. The The dashed curve has been calculated from the theory given in Appendix D. See text p p . 29, 31, 39,41£.,47, 77, 79, 84. .... .... lJ1 I PI I ..... Z <I o17 (p,a) N 14 8LAS =150° o <I a: w ICf) I Z o I- 2 o a: a. I I I ....... /1 Cf) <I :x: a. I ..J <I '0 I I ...... o ..J W ', I I ><I :x: a. ..J <I / In /1 11 ~l'I \ \ I\ ,, \ (JI '1 \ \ \1 \ \ \ II \ <I ..J 925 FIG. 16 930 935 PROTON BOMBARDING ENERGY, LAB (kev) 940 Figure 17. Laboratory alpha-particle yield Y L vS proton bombarding energy E1B near the 11OI-kev narrow resonance. The error bars indicate statistical errors only. The dashed curve has been calculated from the theory given in Appendix D. and the dotted curve is the assumed background from the broad l096-kev resonance. See text pp. 29. 31. 37. 4lff.. I ..... ..... a- 47. 77. 79. 84. '" I -I 16- o ! r 1 • ~ > Q) I .,- 0 z I 0 ~ 0 ~ a.. ,.. N 10 • / .x " / "III >- ~ -0 CD"" (!) ...J • LOa:: -w 1/ / =z / w / (!) z o t' • I • a:: <2: 1 \ • ID \ :E "~ o OlD "- ..... =z ........ I " "- • o ~ o I "" a:: a.. 1-+-1 "I---+.--i , . " \ . \. LO o \' I • I -,.... . (!) lL.. (N'110\1~3.lS-NO.lO~d/S\1Hd1'1 01) 0131A '1Hd1'1 8'11 11- , Figure 18. Laboratory alpha-particle yield Y L vs proton bombarding energy EIB near the l247-kev resonance. error bars indicate statistical errors only. The The dashed curve has been calculated from the theory given in Appendix D. and the dotted curve is the assumed background .... .... ..., from the l274-kev resonance. i>I 47.77.79.84. See text pp. 29. 31. 39. 41£f .• I I ....... z ~ a ~ cr 10 w Ien 0 17 (p,a ) N BLAB =150 I4 0 1 z o Io cr I a.. "en I / /I . . . \ \ / I ~ :I: // I I I I \ ' I ;I ' "- --,.- ' .!. a.. ;; , ..j ...J -~ '0 .1 1 5 1I . .1 · I ./ . / / ...J w >~ :I: a.. ...J ~ CD ~ ...J 1240 FIG.IS 1245 1250 1255 1260 1265 1270 PROTON BOMBARDING ENERGY, LAB (kev) 1275 Figure 19. setting N/ V Alpha-particle counts per unit fluxmeter m corrected for charge exchange vs flux- meter setting V rn . tistical error only. The error bars indicate the staThe data in this figure were .... .... I C):) derived from the uncorrected data in Fig. 5. For this data E1B = 1. 280 Mev and Q = 88.6 fJ.coul for each point. See text p. 30. P> I -I 1 8- It) en v •-Z 0 0 ~ tj -- ~ a... It) • II m .... « 0 Q) ..-! -J 0 0 It) t+t ~ > -E I-+-l t---l <!) t-+-t It) 0 .........-t It) t---l Z -~ ~ W (J) I--+--l a:: I-+-l w ~ 0 t-t It) t+t ~ W ~ X :::> ~ -.J lot lL.. lot It) It) o N It) o It) N o It) o 9NI113S ~3l3WXnl.:l liNn ~3d SlNno~ 'tHdl't a3l~3~~O~ (I_ AW ) (1) ,. • Figure 20 . setting N/V setting V m Alpha-particle counts per unit fluxmeter m corrected for charge exchange VB fluxmeter corrected for carbon build up. indicate statistical error s only. The error bars The data in thi s figure we re derived from the uncorrected data in Fig. 6. The I .... .... '" I» circles. triangle. and square are data taken with different reference profiles. For this data E1B = 854 kev and Q ;: 2850 fJ.coul for each point. See text p. 31. I -I 19 - 10 V •-Z 0 10 0 ~ It) t:S • el- II ...--- m <t ~ ..J 0 CD 0 • 10 10 (!) z ..... ..... • <l W 10 10 10 • • • 0 (/) a: w ..... w ~ X • I > E ...... ::::> 0 CD 10 <l -.J lL.. 0 W • ..... (,) w a: 10 a: CD 0 (,) o I'10 o o o o 10 9N 1113S ~313~Xnl.:i liNn ~3d SlNno:::> '1Hdl'1 a31:::>3~~O:::> ( I_AW) o C\I (!) lL.. -120- 4 .0 61RON OXIDE TARGET 017(p,a)N I4 • NICKEL OXIDE TARGET BLAB = 150 0 1.0 z <{ o <{ a:: w 0.4 ~ I en en ....... z a:: <{ (I) ...J ...J 0.1 ~ (I) <{ ...J 0.04 z o ~ <.> I I w en en 0 .01 en o a:: <.> FIG.21 600 700 800 900 1000 PROTON REACTION ENERGY, LAB (kev) 1100 -120a- Figure 21. IT L Laboratory differential cross section (9 ) vs proton energy El in the energy region where L the measured cross section was lowest. logarithmic scale for the ordinate. Note the The error bars indicate statistical errors only, and the absolute cros s section scale is accurate to 10"10. both types of target are shown. Data from The vertical lines indicate the positions of narrow resonances. text pp. 31, 47. See Figure 22 . Differential cross section .,.(9) in the c.m. system vs. c. m. energy E in the 0 17 + P system. The figure shows the experimental results near the two broad resonances at c.m. energies of 1203 and 1035 kev (lab energies of 1274 and 1096 kev ). indicate statistical errors only. The error bars The dashed curve is that computed by summing two single level formulas, one for each of the two broad resonances . pp. 24, 31, 36, 39, 45, 47 . See text .... N .... III I I T- ~ ---T I Z <t OI7(p,a)N • NICKEL OXIDE TARGET 0 <t a:: ~ IRON w OXIDE TARGET I I- en ....... en z a:: , I 10 I <t ..J ..J z 5 t) ~ t) \ \ \ ~ I 1- --A- 1 .- 950 FIG,22 1- / I 1050 I 1100 II ~~E !-- . \ " " '- 1: 6/ J-l..I ;t" y £ \ I 1/ II # \ ~ I ~~ \I fl I ••• - 'h JJ w en a:: II I l' 0 0 I , , --- en en \\ ~/ ~ t) 8 LAB =150 o ,1 Ii \ ,I CD l- lIi 1: "- I "- ...... _ l! / I 1000 I 1150 I4 I 1200 C. M. ENERGY (kev) I 1250 I 1300 1 -- -- -- I 1350 F\gure 23. p-wave barrier factors r / e2 for both incoming and outgoing channels vs c . m. energy E in the 0 17 + P channel. Points are taken from Tables II and III, and the dashed curves are polynomials fitt e d to these points. See text p. 38. I ..... N N I" I Figure 24. Energy relations in the target used in calculating the energy spread due to the angular acceptance of the spectrometer. p. 72. See Appendix C. I .... N Vol i» I -123- w U <l: lL. 0: :::l (f) '<t N ~ lL. Figure 25. 2 p-wave barrier factor r / e for the N channel vs c. m. energy E in the 0 17 14 + Ci + P channel. Points are taken from Table III and the dashed curve is a polynomial fitted to those points . See text p. 53£. .... N I >l- i» I -124- , 0 0 \ \ ~ \ \ \ \ \ \ .....> \ \ Q) .JI: \ ..\ 0 \ 10 \ >(!) a:: q w Z \ W . \ ~ \ \ U \ \ ...J W \ \ Z Z \ <t \ I U 0 \ \ q t1 •-+Z \ \ \ IC) N (!) lL. 10 0 C\I N ("8)1) 10 0 It? 0 ~Ol.:)'1,:j ~31~~'18 31\'1M-d Figure 26 . in the 0 17 Cross section factor S(E) vs c . m. energy E + P system at the stellar energy region . Both the case of constructive interference and of destructive interference between the levels are shown. At the high .... N I energy resonance S rises to 6.88 x 10 textpp. 54, 56. 6 key-barns. See <.n i>' I -125- - -----:-::=--=--- - - - o ~ 0 --- --- - ID -- - - ....... "- 0 "- It) "\ \ \ 0 v \ \ en z 0 u \ I I I t-= en w I 0 I I ---w > CD .:.t! 0 It) 0 C\I / I I / I / --- - - - - - ./ ...... .; 0 ./ <D C\J (!) u.. 0 0 0 0 0 0 C\I (U J oq --'8lj) ( 3) S Figure 27. E in the 0 17 Folded rate integrand 4> cf(E) vs c. m. energy + P system . The area under this curve is proportional to the 0 17 (P. Q!)~4 reaction rate at a temper- ature of 15 million degrees. This curve is for construc- tive interference between the two levels which contribute .....I N to the stellar cross section. See text p. 60. ali> 1 -126- o r-- ~ 0 "'0 0 \D u; x 0Z 10 _u II 0 ..... It) ~ > 0 V CI) ~ ~ W oIt) o . (!) lL.. It) '" It) r-- oIt) It) '" Figure 28. Folded rate integrand II>cf{E) vs c.m. energy E in the 0 17 + P system . Part of the unfolded integrand II> (E) is shown as a dashed curve. The area under this c 17 14 curve is proportional to the 0 {p, 0!)N reaction rate at a temperature of 15 million degrees. This curve is for destructive inte rference between the two levels which contribute to the stellar cross section. See text p. 60 . I ..... N ...., P> I -127- 0 III 0 ,... 0 ...J 0 LL Z / - '\ ~ 0 ::::> II) 0....,: X 0 ID Cf) W 10 0 II I- 0 LO o > Q) V.Jt: ~ w o C\I o Figure 29. E in the 0 17 Folded rate integrand <I> citE) vs c. m. energy + P system. Part of the unfolded integrand <I> (E) is shown as a dashed curve. c The area under this curve is proportional to the 017(p, a)N 14 a temperature of 20 million degrees. reaction rate at This curve is for .... N I constructive interference between the two levels which 00 i» contribute to the stellar cross section. See text p. 60. -128- 0 ILl 0 -.J 0 ,.... 0 iL/"_ z " y: ::::> 0 ID , I o _enXz 0 ID 00 NU II 0 t- 10 o o,.... o ID o 10 o V o If) o (\j o Figure 30. E in the 0 17 Folded rate integrand <I>cf(E) vs c.m. energy + P system. Part of the unfolded integrand <I> (E) is shown as a dashed curve. c curve is proportional to the 0 17 The area under this 14 (P, 0!)N a temperature of 20 million degrees. reaction rate at This curve is for destructive interference between the two levels which I I-' N contribute to the stellar cross section. See text p. 60. '"'Il> I - 129 - o w o o I'- -l o lL. -- - - -- . / - Z :::) "- o ID ) / /' /' '" ./ o lO w (UJoq-Aa)! 01) (3)i~~ 9Z- '%' . Flgure 31. Log(O17 P16 / 0 16 P 17 ) vs temperature T as cal- culated in the text and given in Table XII. Both the case of constructive and of destructive interference are shown. The dashed curve is what one obtains on assuming the entire 0 (51). 17 (p, 0!)N 14 .....I rate is given by the resonant formula Eq. At equilibrium we have P '" o = P 17 , and the ordinate 16 then gives the logarithm of the concentration ratio. III I See text p. 6lf. • I \ \ \4 +5 RESONANT CONTRI BUT ION TO 0 17 (p,a) N I4 \ \ \ \ \ ........-;--.. 10 \ \ ~ t- o \ Il. 1 Il. DEST. lID l>I o \ \ °1° '-.:../ 0 - - - - - - - - - - ~- - - -----~-- (!) ° ~\- - - - - - - _ ..)...~~ - - - - - - - - 0.170} = [ 0 .072 17 o /0 16J CNO ~ ...J CONS. -5 5 FIG .31 10 15 20 25 30 35 6 TEMPERATURE (10 OK) 40 45 Figure 32. A comparison of the true rate correction P c with the rate Po as calculated by the nonresonant approximation (NRA) vs temperature. Both constructive and de- structive interference are shown. The vertical arrow indicates the temperature at which Po = O. See text p. 60. .....I ""..... III I -I 3 1- 0 en z 0 u 0 If) I- en w ~ 0 <] 0 <] It) ~ C\I 0 <II 0 ~ <] 0 0 C\I . :::J I<] It) 0 ~ 0 <] It) It) o o N (o'd) 0 . • 1 LI ct:: w <t> o c::x: a.. <]0 It) W ct:: ~O.:l (~ddH)Ol y w I-