A Study of T=2 States in ¹²B, ¹²C, ²⁰F and ²⁸Al Citation Nettles, Patrick Henly, Jr. (1971) A Study of T=2 States in ¹²B, ¹²C, ²⁰F and ²⁸Al. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/SC5V-NJ66. https://resolver.caltech.edu/CaltechTHESIS:09222015-134005716 Abstract The lowest T = 2 states have been identified and studied in the nuclei 12 C, 12 B, 20 F and and 28 Al. The first two of these were produced in the reactions 14 C(p,t) 12 C and 14 C (p, 3 He) 12 B, at 50.5 and 63.4 MeV incident proton energy respectively, at the Oak Ridge National Laboratory. The T = 2 states in 20 F and 28 Al were observed in ( 3 He,p) reactions at 12-MeV incident energy, with the Caltech Tandem accelerator. The results for the four nuclei studied are summarized below: (1) 12 C: the lowest T = 2 state was located at an excitation energy of 27595 ± 20 keV, and has a width less than 35 keV. (2) 12 B: the lowest T = 2 state was found at an excitation energy of 12710 ± 20 keV. The width was determined to be less than 54 keV and the spin and parity were confirmed to be 0 + . A second 12 B state (or doublet) was observed at an excitation energy of 14860 ± 30 keV with a width (if the group corresponds to a single state) of 226 ± 30 keV. (3) 20 F: the lowest T = 2 state was observed at an excitation of 6513 ± 5 keV; the spin and parity were confirmed to be 0 + . A second state, tentatively identified as T = 2 from the level spacing, was located at 8210 ± 6 keV. (4) 28 Al: the lowest T = 2 state was identified at an excitation of 5997 ± 6 keV; the spin and parity were confirmed to be 0 + . A second state at an excitation energy of 7491 ± 11 keV is tentatively identified as T = 2, with a corresponding (tentative) spin and parity assignment J π = 2 + . The results of the present work and the other known masses of T = 2 states and nuclei for 8 ≤ A ≤ 28 are summarized, and massequation coefficients have been extracted for these multiplets. These coefficients were compared with those from T = 1 multiplets, and then used to predict the mass and stability of each of the unobserved members of the T = 2 multiplets. 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): Barnes, Charles A. Thesis Committee: Unknown, Unknown Defense Date: 9 October 1970 Funders: Funding Agency Grant Number National Defense Education Act UNSPECIFIED Office of Naval Research UNSPECIFIED NSF UNSPECIFIED Caltech UNSPECIFIED Record Number: CaltechTHESIS:09222015-134005716 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09222015-134005716 DOI: 10.7907/SC5V-NJ66 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9167 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 22 Sep 2015 21:55 Last Modified: 24 Jun 2024 20:45 Thesis Files Preview PDF - Final Version See Usage Policy. 27MB Repository Staff Only: item control page

A STUDY OF T = 2 STATES IN l~, 12 c, 20F AND Thesis by Patrick Henly Nettl es , Jr . In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, Cal ifornia 1971 Submitted October 9, 1970 20Al ii ACKNOWLEIXH'1ENTS It has been a distinct pleasure to work in the Kellogg Radiation Laboratory, and I am deeply indebted to all of the staff members for their interest and support in this work. I would especially like to thank Dr. C. A. Barnes, who has guided this program, for many fruitful suggestions and enlightening discussions. Much of this work was done in collaboration with Dr. D. c. Hensley. Sincere thanks are extended to him for introducing me to these studies and for education in the laboratory. The assistance of Dr. D. B. Nichols and P. Dyer in the later part of the work at Caltech is appreciated. The experiments at Oak Ridge were done in collaboration with Dr. C. D. Goodman. His assistance in arranging beam-time for these experiments and in planning and executing the measurements is gratef'ully acknowledged. Financial assistance was provided under the National Defense Education Act (Title IV), and by the Office of Naval Research, the National Science Foundation, and the Caltech Physics Department. Finally, I would like to thank my wife, Cathy, for her patience and support through the many long days and late nights while this work was in progress. jij_ ABS'rl'tACT The lowest T = 2 states have been identified and studied in . 12 12-- 20 28 the. nuclei c, -.s, F and Al. produced in the reactions The first two of these were 12 14 14 3 1 c(p,t) c and c(p, He) 2:s, at 50.5 and 63.4 MeV incident proton energy respectively, at the Oak Ridge 2 28 National Laboratory. The T = 2 states in °F and Al were observed in ( 3He,p) re~ctions at 12-MeV incident energy, with the Caltech Tandem accelerator. The results for the four nuclei studied are si.lmmarized below: (l) 12 C: the lowest T = 2 state was located at an excitation energy of 27595 ± 20 keV, and has a width less than 35 keV. (2) 1 ~: the lowest T = 2 state was found at an excitation energy of 12710 ± 20 keV. The width was determined to be less than 54 keV and the spin and parity were confirmed to be o+. 1 A second ~ state (or doublet) was observed at an excitation energy of 14860 ± 30 keV with a width (if the group corresponds to a single state) of 226 ± 30 keV. (3) 2 °F: the lowest T = 2 state was observed at an excitation of 6513 ± 5 keV; the spin and parity were confirmed to be o+. A second state, tentatively identified as T = 2 from the level spacing, was located at 8210 ± 6 keV. (4) 28 Al: the lowest T = 2 state was identified at an excitation of 5997 ± 6 keV; the spin and parity were confirmed to be o+. A second state at an excitation energy of 7491 ± 11 k eV is tentatively identified iv as T = 2, with a corresponding (tentative) spin and parity assignment J~ 1. The results of the present work and the other known masses of T = 2 states and nuclei for 8 < A < 28 are SUilllllarized, and massequation coefficients have been extracted for these multiplets. These coefficients were compared with those from T = 1 multiplets, and then used to predict the mass and stability of each of the unobserved members of the T = 2 multiplets. v TABI,E OF CONTENTS ll'ITLE PART PAGE I. GENERAL INTRODUCTION l II. EXPERIMENTAL METHOD 3 A. Introduction 3 B. Target Preparation 4 C. Determination of Target Thickness 7 D. Magnetic Analysis 9 E. Particle Detection and Position Measurement 2 STATES III. INTRODUCTION TO T IV. THE LOWEST T = 2 STATE IN 12 1 c AND 2:s ll 13 19 A. Introduction 19 B. 14 12 The Reaction c(p,t) c 22 1. Experiment 22 2. Excitation Energy 25 3. Angular Distribution 28 4. Width 29 5. C. D. Conclusions 1 14 3 The Reaction c(p, He) 2:s 30 l. Experiment 3l 2. Excitation Energy 34 3. Angular Distribution 35 4. Width 36 5. Conclusions 37 Prediction of the Mass of 1 2:se 31 38 vi PAR'r v. VI. Pl\m: 2 T = 2 STATES IN 0F A. Introduction B. Experiment 42 C. Excitation Energy 43 D. Angular Distribution 46 E. Conclusions 47 28 T = 2 STATES IN Al A. Introduction B. Experiment C. Excitation Energy D. Angular Distribution E. VII. VIII. Conclusions 41 41 48 48 49 5l 53 54 T = 2 MULTIPLETS AND THE QUADRATIC MASS EQUATION A. Introduction 56 B. Mass Equation Systematics for T = 2 Multiplets 57 C. D. Predictions for Unobserved Proton-Rich Members Conclusions 59 CONCLUSION 56 60 62 PARTICLE ENERGY DETERMINATION FROM SPECTROGRAPH DATA 65 APPENDIX B• . POSITION-SENSITIVE DETECTORS 69 APPENDIX C. ANALYSIS OF ANGULAR DISTRIBUTIONS FOR TWO-NUCLEON-TRANSFER REACTIONS 72 THE QUADRATIC MASS FORMULA; COULOMB ENERGIES; AND MASS PREDICTIONS 76 APPENDIX A. APPENDIX D. REFERENCES 78 vii PART •rrrr..Jt.: TABLES 83 FIGURES l02 viii LIST OF TABLES TABLES TITLE 1 Fractional Changes in Spectrometer Calibration 2 3 4 5 PAGE Factor with Magnetic Field 83 Excitation Energies Predicted for the Lowest 12 1 T = 2 States in c and 2:s 84 14 c Target Thickness/Composition Measurements 3 from 12-MeV He Scattering at 90° 85 Measurement of the Excitation Energy of the 12 Lowest T = 2 State in c 86 Width Analysis for 14 12 14 11 c(p,t) c and c(p,a) B Measurements 88 1 2.s States 89 c(p,~e) 12.s Measurements 91 6 Measurements of Excitation Energies of 7 Width Analysis for the 8 Predictions of the Mass of 9 Predictions and Previous Measurements of the 14 1 2:se 92 Excitation Energy of the Lowest T = 2 State 10 11 in 13 93 94 Predictions and Previous Measurements of the 28 Measurements of the Excitation Energ ies of States 28 in Al Al 95 96 Optical Potentials Used for Analysis of Angular Distributions 14 °F Measurements of the Excitation Energy of T = 2 2 States in °F Excitation Energy of the Lowest T = 2 State of 12 2 Sunnnary of Masses :ear the Lowest T in A = 4n Nuclei 97 2 Multiplets 9U ix TABI.F~G Tl'l'Ll!i l5 Mass-Equation Coefficient s and Coulomb Ener g ies for A = 4n, T = 2 Multiplets 99 16 Mass and Stability Predictions for T = -2 Nuclei z 100 17 Mass,. Excitation Energy and Stability Predi ctions for the Lowest T = 2 States in A. = 4n, Tz = -l Nuclei lOl 1 I. GENERAL INTRODUCTION From a variety of phenomena observed in various facets of nuclear physics and particle physics, it has been found that the strong (nuclear) forces are charge -independent to within about ~ (Henley 1969). This fact is manifested, among other ways, in the existence of isobaric analogue states, i.e., states which have the same structure in each isobar of a particular mass -number, except that the total nuclear charge is 0.ifferent from one member to another. The principal charge-dependent forces -- the Coulomb and spin-orbit interactions -- give rise to a mass splitting for these states, which is easily predicted from perturbation theory, at least in principle . The studies to be described in the present work are part of a continuing program at Caltech, and a considerable effort at many other laboratories, to measure the mass splitting for these multiplets. The experimental techniques involved in this particular work are outlined in the following chapter. In Chapter III the previous work from the Caltech progrrun is summarized and the goals of the program are discussed in more detail . concepts used in this work are introduced . Also, some elementary Chapter IV discusses two experiments performed at Oak Ridge, using protons from the Isochronous Cyclotron to excite two members of the lowest isospin quintet in mass 12. Chapters V and VI deal with 3 He -induced reactions studied at Caltech, which excited one member of each of the lowest quintet in A= 20 and 28, respectively. In Ch apte r VII 2 a survey of the available measurements for isospin quintets in light ' nuclei with A = 4n is compared with the mass formula predicted f rom first-order perturbation theory, and the masses and decay modes of the unobserved members of these multiplets are discussed . 3 A. Introduction The experiments discussed in the following chapters have a number of experimental techniques in common. In all of these experiments, charged-particle reaction products were identified by combining energy, momentum/charge, and energy-loss measurements. These were obtained by first momentum-analyzing the reaction products at a known laboratory angle in a magnetic spectrometer, then allowing these particles to penetrate a foil of knm.nthickness, and finally measuring the resulting particle energy spectrum in a solidstate detector (Figure 1). Generally, the reaction was observed in transmission geometry, where the incident beam enters one side of the target and the reaction product to be observed exits through the opposite side, which required that the targets be r easonably thin (i.e., the fractional energy loss for both beam and reaction products should be small). Then from a knowledge of the target thickness obtained from energy loss or resolution measurements and a measurement of the magnetic field and focal-plane position, the exiting particle energy b efore leaving the target could be determined precisely. By comparing the energy of a particle group of interest to that of one or more groups from the same target and beam with precisely known Q-value(s), the Q-value for the reaction of interest was determined. Most of the details of this procedure and the analysis of the pro1.Ja1.Jle e rrors incurred have been discussed 4 previously by Hensley (1969). B. Target Preparation To meet the requirements outlined above, the targets used in these experiments were prepared either as thin self-supporting foils of more or less uniform composition or as multilayer foils w.i.th one layer of target material and the remaining layer(s) for support. The supporting foils for most of the work were ~100 to 300 µg/cm gold, evaporated by the standard techniques in vacuum. 2 For most of the experiments, the target materials were (relatively expensive) isotopically-enriched substances, so the preparation procedures were required to be reasonably efficient, to avoid unnecessary waste . For the experiments done at Caltech, foils were mounted on the customary 10-mil tantalum frames over a 5/16 - inch-diameter hole. The targets used at Oak Ridge were mounted on a 15-mil tantalum frame adapted to fit the standard target holders in use there. These targets were mounted over a 1/2-inch diameter hole to allow for the larger beam area from the cyclotron. (l) 26 Mg Targets . MgO, enriched to 99% in The 26 Mg targets were prepared by reducing 26 Mg, under vacuum, by heating a mixture of MgO and Ta powders in a carbon boat . To improve the efficiency of the evaporation, the boat was made in the form of a vertical cannon, af'ter a design by Goosman (1970) . Magnesium freed by the reduction process at the bottom of the cannon was then confined to a relatively 26 small cone above the source. This Mg vapor was condensed on a thin gold foil mounted on a target frame and suspended about 2 inches I ' •.J above the cannon. Carbon-backed Mg targets used in early experiments were prepared in a slightly different manner. In the procedure outlined above, it was found that the Mg vapor would not condense satisfac2 torily on a thin ( 20 - 100 µg/cm ) rounted carbon foil. However, the vapor was easily collected on a carbon foil before it was removed from the glass slide. The foil was then cut into suitable squares , floated from the slide and successf'ully mounted. This process allowed the collection of a larger fraction of the released vapor; however, it was still a rather unsatisfactory technique since thin Mg layers deteriorate quickly by chemicaJ_ reaction with water (including water vapor). The appearance of the targets was notice- ably different near the edges a~er a few minutes in the floating dish. No detailed investigation of the quaJ_ity of the targets was made, so the results are uncertain. The same technique was attempted with gold foils; in that case, pinholes in the gold foil allowed direct contact between the water and the Mg, with disastrous results. In the evaporation process, it was found that the MgO apparently released a considerable amount of absorbed gases -when initially heated, until the boat was bright red. Unless the temperature was increased very slowly, the powder in the cannon had a strong tendency to jump out. To avoid this annoyance, the design of the cannon was modifie d slightly. The top of the boat was extended above the electrode about 0.3 inch so that a horizontal copper cover-plate, which could be moved from outside the vacuum system, would cover and nearly touch the top of the boat. With this cover in place, the 6 powll.er could 1)(1 heated rapidly wlth01it any r:qJpree:Lal>l.c :Lunn or MgO. (2) lllo Targets. Solid oxygen to.rgctu were prepared liy oxidizing thin nickel foils, by heating them in an oxygen atmosphere w.ith a projection la.mp. Since the commercial nickel foils were prepared w.ith a thick copper backing for easier handling, the foils were generally first mounted on the frames w.ith silver print or epoxy, after 'Which the copper backing was etched away chemically w.ith a mixture of ammonium hydroxide and trichloroacetic acid (Richards 1960). In this procedure, the thinner nickel foils became very tightly stretched on the frame, and they consequently showed a strong tendency to break or split as they were heated. To improve the yield of useable targets, a special foil holder was devised which shielded the target frames from the direct heat of the lamp, allowing only the central 1/4-inch diameter of the foil to be heated. With this simple device a considerable reduction was achieved in the number of foils lost during the oxidation process. 0 For the thinnest (500 A) nickel foils, a further improvement in yield was desirable. Before mounting the foils, the copper back- ings were removed by floating the foil on the surface of a pool of the etching solution. The bare nickel foil was then li~ed out of the etch on a glass slide and place d in a pool of distilled wate r. This wash step was repeated several times, li~ed from the surface on a target frame. a~er which the foil was Foils prepared in this way were considerably looser on the frame, and were much easier to oxidize 'Without breakage. 7 (3) l4 C Targets. The process for preparing l4 C targets was developed jointly with Hensley (1969), from an idea of Douglas (1956). 14 An A.C. discharge was established in c-enriched acetylene between two electrodes separated by ~o.5 cm. The acetylene deposited as a polymer on foils mounted on the electrode surfaces. The foils con2 sisted of 300-µg/cm Au with a backing of 0.1-mil connnercial rolledcopper foil. After the polymerized-acetylene layer was deposited, the foils were mounted on Ta frames and the copper was etched away chemically. For the thicker targets used in the experiments described in this thesis, an additional thin layer of gold was evaporated over the carbon layer to provide better electric8.l conductivity. In the deposition of thick layers (of the order 0.5 mg/cm 2 ) it was found that the discharge of'ten produced sparks or hot spots on the target surface. Subsequent examination of. the surface in- dicated that these sparks were associated with flaws such as pinholes in the deposited l ayer, which offered a path of lower resistance for the discharge. To retard the aggravation of flaws by this hot-spot phenomenon, a current-limiting resistor was introduced between the tesla coil and electrode, of the order of lOO Kn. With this modification, there was a noticeable reduction in hot spots, and it was possible to prepare reasonably uniform targets of the required thickness. c. Determination of Target Thickness For solid targets, the thickness can be determined from yield 8 comparisons, energy-loss measurements, or resolut:l.on meu.surc!'lricmtn as illustrated below. (l) Magnesium targets. The thicknesses of the magnesium targets used for this work were measured by observing elastically scattered 3 He from the gold (or carbon) backing, both directly and through the Mg layer, in the spectrometer at 90° in the lab (Figure 2). The inverse process (scattering from the Mg) was used to measure the thickness of the backing (Figure 3) • .An additional measurement of the thickness was obtained from the energy-width of the elastically scattered particle group when the target thickness was greater than the instrumental resolution. (2) NiO Targets. 18 The thickness of the Ni o targets was determined from the nominal Ni-foil thickness and a comparison of 3 the yields from the ( He,a) reaction on the Ni pure 18 o, in a gas cell. 18 o target and on Additional thickness information was extracted from the width of elastic scattering groups from 58 Ni and lBO. (3) 14 c targets. The 14 C target material was enclosed between two layers of gold, one of which was considerably thicker than the other. The thickness of the polymer layer (assumed to be entirely carbon, for energy-loss calculations) was measured by observing the elastic scattering of 12-MeV 3 He from the gold layers, in the magnetic spectrometer at 90° in the lab (Figure 4). The target thickness was extracted from the energy difference oi' the two pea.ks corresponding to the two gold layers. The thiclmess of 9 the thicker gold layer was determined from the width of the directlyobserved elastic scattering peak. This thickness was scaled by the ratio of yields to give the thickness of the thinner gold layer. D. Magnetic Analysis The separation of charged particles by magnetic analysis is an established technique for nuclear spectroscopy, in use in many laboratories. It offers the advantages of excellent .particle identification (especially when combined with an energy measurement at the focal plane), and lower count rates (by selecting for observation only the smal.l portion of the particle spectrum of interest). These are gained at the expense of solid angle and field of yiew in the particle momentum spectrum; the technique is best applied to precise measurement of a particle spectrum over a reasonably narrow and well-defined range of energy. In all the experiments under consideration in this work, charge·d -particle reaction products were observed ·in either the 61-cm-radius, double-focusing magnetic spectrometer at C8.J..tech, or the broad~range magnetic spectrograph at the Oak Ridge cyclotron laboratory. The Caltech spectrometer has been discussed in detail by Groce (1963), McNally (1966), and Moss (1968 ), and some details of importance for the present work were considered by Hensley (1969). The expression used to determine particle energy from the measured NMR frequency is presented in Appendix A for this instrument. The broad-range spectrograph at Oak Ridge is patterned in 10 principle af'ter a design by Borggreen et al (1963), which differs from that of the Caltech double-focusing spectrometer in several ways. The usable focal plane is 2 meters long, spanning a large range in momentum (for the experiments discussed here, this design feature was not utilized). Since it is a uniform-field magnet, the spectrograph has focusing properties in only one dimension, aside from the small effect of the fringing field at the edge of the magnet. The instrument is oriented horizontally so that the focusing occurs along the direction of the kinematic angle for a reaction. By selecting the position of the moveable focal plane correctly for a particular reaction, the focusing can be made to cancel (approximately) the peak broadening associated with the kinematic energy shif't across the aperture, allowing measurements to be made with experimental resolution considerably better than the kinematic shift over the aperture. The orientation of the focal plane for this instrument is thus a critical parameter, since improper positioning introduces excessive broadening for the observed particle groups. Since the reactions studied with this instrument involved light nuclei which give large angle-dependent kinematic broadening, the focal-plane position was reset af'ter each change in angle. With the focal plane adjusted in this way, the resolution (calculated by a computer code at Oak Ridge) was domi nated by the effect of the last quadrupole focusing magnet before the scattering chamber, for the (p,t) reaction, and by the target 3 thickness, for the (p, He) study. The orientation of the focal plane enters the calculation of ll particle energy from the measured position a long the focal plane . This calculation is discusoed in deta:ll in J\:ppencl:lx A. Since there is essentially no i'ocusing in the other ungitlo.r dimension for this instrument, the effective solid angle for an observation depends on the position along the focal plane and on the height of the detector. where 6D. This relationship is given by (Ball 1968) is the solid angle in msr, R the average orbit radius in inches (Appendix A), h the detector height in inches, and 6Q the total angul~r opening of the spectrometer entrance slits along the kinematic angle, in degrees . E. The coefficients C are: n n cn 0 0.88611 l - 3 .6023 x lo- 2 2 5.653 x l0- 4 3 - 3 .116 x 10-6 Particle Detection and Position Measurements To determine the energy of a particle group observed in a magnetic analyzer precisely, it is necessary to measure the location of the group on the focal plane precise ly. For the experiments with the Caltech spectrometer, this was achieved through the use of a 16-counter array spanning the focal plane or a position-sensitive solid-stute deteei~or (~5 cm long). The rnen::.;1.u·c1111.mtu nt Ouk H:ldge were made with two position-sensitive detectors on the focal plane . The design and operation of the l6-counter array has been discussed by McNally (1966) and Moss (1968). HensJey (1 969) has discussed the details of peak profiles and yiel d measurements with this system, which apply to the present work . The operation and calibration of position-sensitive particle detectors is discussed in some detail in Appendix B. The associated electronic circuits used for these detectors are shown schematically in Figure 6 (Caltech) and Figure 7 (Oak Ridge) . also discussed in Appendix B. These systems are 13 INTRODUCTION TO T = 2 STATES III. The study of T = 2 states in Tz = 0 or Tz = 1 light nuclei is a natural extension of the earlier studies of T = 3/2 states in T = ± 1/2 z nuclei in this laboratory (Lynch 1965, Dietrich 1965, Adelberger 1967, Hensley 1969 and McDonald 1969). In most cases the T = 3/2 or T = 2 states were populated as final states in an isospin-allowed reaction, 3 3 ( He,n) (Adelberger 1969a, 1970) or ( He,p) (Hensley 1968 and the 1 present work), on T = 2 or T = 1 targets. Because of the general interest in the T = 2 states in mass 12 and the difficulty in obtaining a highly enriched 10 3 3 Be target for ( He,p) and ( He,n) studies, 14 3 the (p,t) and (p, He) reactions on c described in this thesis were included in the program. The principal goals of the program have been (1) identification of the T = 2 levels and (2) precise measurement of the reaction Q-values to extract accurate excitation energies for these levels. The strong interest in measuring the excitations accurately is motivated by the continuing need for a more reliable mass formula, to predict masses of nuclei far from the line of stability for astrophysical as well as nuclear purposes. The masses of such nuclei are closely related to those of the analogue excited states and the latter are frequently more easily produced in the laboratory. Thus the program was e ~tablished to produce precision measurements by which any proposed mass formula or relationship could be tested in detail. The i dentification o:f an energy level as a.n analogue st11te is ne ce s sarily o.n indi re ct proce s s . The t'eat11rt)S which huve l"._'t' t1 u ~:e 1'11]. 14 in identif'yin.g these states are the ir predicte d Gpins and paritie s, e x citation ene rgies, widths and the ir tro.no i ti on otrcnKtlw in various reactions. 3 3 The reactions ( He,p), (p,t), and (p, He) are assumed to proceed by tne direct transfer of a T = 1, J~ = O+ di-nucleon, to populate the levels of interest. The selection rules for these reactions are discussed in Appendix C. The T 2 states studied in this work were all members of multiplets with even-even parent nuclei; the spin and parity are therefore expected to be o+, from nuclear systematics. To populate such a level in one of the above reactions on a (T = 1) o+ target, the L-value for the transferred particles must be zero, uniquely. forward~angle This generally gives the characteristic peaking for the differential cross section, if the state is populated by a direct reaction. Since the reactions in which the T = 2 states were excited are allowed by isospin selection rules, it is expected that the measured cross sections should be larger than for similar processes that are forbidden by these rules. Thus a T = 2 state should be absent or only very weakly excited by an (a,d) reaction on a T = 1 3 target, while it should be strongly excited by a ( He,p) reaction on the same target. Such comparisons are one-way devices however, since the effects of structure and reaction mechanism are still to be accounted for. Thus the strong excitation of a suspected T = 2 level as the final state of an isospin-forbidden reacti on is good evidence that it is not T = 2, while the absence of the level in an isospin-allowed reaction is of little significance (without 15 adcll tional information or aus111Jlpt:lono au out tlte GtrHcturc) . Dpce:U.':lc examples of the effects of structure on transition strength have l.Jeen 3 discussed by Adelberger (1967) in the case of ( He,n) versus (p,n) 3 . 3 reactions, and HensJey (1969) in the case of ( He,a) versus ( He,p) reactions. In the A= 4n nuclei, the T = 2 states are described as two-particle, two-hole states. These should be excited easily by the isospin-allowed two-nucleon transfer reactions under consideration. The excitation energy for a T = 2 state can be predicted approximately' if the mass of the parent nucleus is known, by assuming the mass splitting of a multiplet to be mainly the result of the neutron-proton mass difference and the repulsive Coulomb interaction between the protons. It is assumed in addition that the repulsive interaction gives rise to a mass splitting of the form KZ(Z - 1) where K is constant across a multiplet. The constant K can be evaluated approximately from the ground-state mass difference for the T and T z = -1 members of the multiplet. = +l z This gives the following results for the excitation energies E (T,T ) in nuclei of mass x z . number A: E)2,l) = [m(2,2) - m(l,l)] + 2(~=l) [ m(~ -l) - m(l,l)]- A:l 6mnp (III-1) and E (2,0) . x [ m(2,2) - m(o,o)J + !:{ [m(l,-1) - m(l,l )] - A~l 6mnp (III-2) 16 where m(T~T ) is the mass excess of the mass-A nuclide wi th isospin z T and z-projection Tz(Tz = N;Z = ~ - Z), and6mnp is the neutronhydrogen mass difference. From the difference of these two equations, we can write a useful relationship: (III-3) Although they are naive in concept, these formulas have been quite useful in selecting the region of excitation energy to be examined in a search for T = 2 states in nuclides for which the parent (T z = +2 ) mass has been measured. In most cases studied in this laboratory, these predictions have been accurate within (roughly) 100 keV. A more general prediction is contained in the familiar quadratic mass law for isobaric multiplets (Wigne r 1957, Weinberg and Treiman 1958 , .Wilkinson 1964a, b, c): 2 m(T,T . z ) = a(A,T) + b(A,T)T z + c(A,T)T z . (III-4) This formula can be derived under the assumptions (1) that the T-nonconserving interactions are sufficiently weak that first-order perturbation theory is adequate for calculating the energy shifts and (2) that these inte ractions are (at most) quadratic in isospinspace (i.e., of tensorial rank 2 or less). The principal T-non- conserving interaction is, of course, the Coulomb force , for whi ch these assumptions are justified. Some additional discussion of this 17 is included in Appendix D. The formulas presented above were derived from a special case of' this formula. From the asoumpt ions outlined (that the Coulomb corrections can be expressed us Z (Z - 1 )K - Z 6m np the coe fficie nts b and c were evaluated approximately in t erms of' mass differences 1 b = ~ [m(l,-1) - m(l,l)] (III-5) and 1 c 2 (A-l)[m(l,-l) - m(l,l) + 6m A~l By combining a Coulomb energy sum rule with the nuclidic mass relationship of' Garvey and Kelson (Ga r vey 1 966, Kelson 1966, Garvey 1969), and considering the special case of T = 2 states i n T = 0 nuclei, Jgnecke (1967,1969) has predicted a relationship between excitation energies Ex(A,T,Tz) Ex (A,2,0) =Ex(A - 1, 3/2, ± l/2) +Ex(A+ 1, 3/2, ± 1/2) (III-6) This rule is expected to be les s accurate than the mass relationship since excitation energies should be more sensitive to details of nuclear structure than Coulomb-energy differences. Adelberger (1970) has tested this prediction for A = 4n nuclei from A= 1 2 to 32, and found discrepancies ranging from 30 keV to 339 keV . From the excitation energies predicted by the s chemes outlined above, for A= 4n (n > 1), T == 0 or 1 nuclei, it is expected that the T = 2 levels are bound (or only slightly unbound, as in the case of 12 c and 1 ~) with respect to isospin-conserving heavy- lB particle decay. 28 2 In many of these nuclei ( °F and .AJ.. are exceptions) there are energetically allowed isospin-nonconse rving parti cle decays. In these cases, the small width of the T = 2 levels, reflecting the isospin inhibition in the energetically allowed decay channels, is as additional characte ristic to b e used for identification. case of 12 C (and 16 In the . 0 ) Adelb erger (1970) has noted that the energeti- cally allowed and isospin-allowed decay by diproton emission is severly inhibited by a small penetration factor, so that the width should remain small compared with the widths of neighboring (T < 2 ) states. 19 IV. THE LOWEST T = 2 STATE IN A. l" '-c AND l') ' 13 Introduction In a review article, Cerny (1968) presented excitation energies for the lowest T = 2 states in · the and l2-:s, ob serve d in 12 14 c, observed in the c(p,t) reaction, 14 3 c(p, He) react•ion. The unce rt aint·ies quoted for these excitation energies (lOO keV and 70 keV, respectively) were considerably larger than the limits desired for the present program. In addition, the data from these experiments have not been published, although the measurements were made several years agoo For these reasons, a decision was made in this laboratory to repeat these measurements (with 14 c targets prepared previously for other experiments) using a magnetic spectrograph to aid in particle identification and to improve the energy resolution. The excitation energy of the lowest T = 2 state in 12 c can be predicted from relation III-6, proposed by J!inecke, or from relation III-2 using either the upper limit for the 1 2:se mass (from ~-decay endpoint energy estimates) (Poskanzer 1965 ) or the 1 2:se mass predicted by the Garvey-Kelson mass formula (Garvey 1966). The results of these predictions are sunnnarized in Table 2, along with the corresponding prediction for the T = 2 level in 1 2i3, from III-3, and the pertinent references. For convenience, the experimental results reported by Cerny are also included in this table. The Q-values for the reactions in question are probably the most negative of any reaction yet studied in nuclear physics (Q ~ -~51 MeV). Thus the bomllarJing energy rm1u:i.red to .1n:i t;J 1rLl.~ ·L11l~ reactions is for beyond the maximum available f'rom the Cu.ltech tandem accelerator, and the measurements had to be made at another laboratory which offered a higher-energy proton beam and a spectro graph . In addition, photographic plates could not distinguish the particles of interest well enough from other particles that were expected to be present, so the spectrograph should be equipped with adequate detectors, at the focal plane . A first atte~pt at these measurements was made by Barnes in collaboration with workers at the University of Minnesota using the proton linear accelerator . (Ol sen 1968 ), For this work the bombarding energy was 40 MeV, and an array of 32, l OOO- µ, surface-barrier detectors was employed for particle detection at the focal plane of the spectrograph. No groups were seen in the regions of interest in either the tri ton or 3 He spectrum at 30° in the lab, but the results were inconclusive because of serious problems in particle identifi cation. Iri return for assistance from the Caltech group in the production of a set of 14 C targets for another experiment of interest to physicists at Oak Ridge, machine time on the Oak Ridge Isochronous cyclotron was made available for a second attempt to find the T = 2 states of interest, with the new and thicker targets . These measure - ments were also made at 40-MeV incident proton energy. Particles from the reaction were observed at the focal plane of the broad- range spectrograph by two solid- state position-sensitive detectors, 5 cm in length . This run wns plugHed 1iy t.l nunillor of' prol>lernc, some 2l associated with the experimenters' unfamiliarity with the laboratory, and some attributed to instrumentation difficulties. Though a considerable amount of time was lost, the (p,t) reaction was studJed at 30° , over a wide range of excitation ener gies centered at the value predicted for the T = 2 state in 12 c. Again, no groups were seen above the continuum yield and background. However, some particle- identification problems persisted, so that the conclusions still remained uncertain. From observations of lower excited states during this run, it was obvious that particle identification was considerably improved at higher triton energy. Oak Ridge, Thus another run was scheduled at to be made at 50 MeV bombarding energy. Again, the (p,t) reaction was studied in detail , and finally a narrow (r ~ 50 keV) group was located. Again a survey was made over several MeV in excitation energy. The data from this run will be discussed in detail in Section B of this chapter. 3 A third run was scheduled at Oak Ridge to examine the (p, He) reaction. In sharp contrast with the two previous occasions, this run was quite straightforward, partly as a result of the experience gained from the previous work, and a lso because the particle identification was considerably improved for 3 He particles. The bombarding energy was chosen to be 63 MeV, to improve the yield of the reaction, and the particles were again observed by position-sensitive detectors at the spectrograph focal plane. corresponding to states in tation from 10 to 15 MeV. C of th:Lu eh11}Tter. 1 Two reasonably narrow groups 2:s were observed in the region of exci- These me asurements are discussed in Section 22 The final section of this chapter is concerned with the prediction of the mass of l2se from the results of these experiments. B. The Reaction l. l4 12 C(p,t) C Experiment . For the final measurements of the reaction 14 C(p,t) 12 C, the beam energy determined from the l53° analyzing magnet was 50 . 535 MeV . The two position-sensitive detectors were placed approximately midway along the focal plane of the Oak Ridge broad-range spectrograph . Both detectors were covered with foils consisting of 3-mil aluminum plus l-mil mylar. The first observations were made to establish the identification of the tritons in the energy spectra from the detectors. To this end, particles produced by bombarding an aluminum foil were observed at a magnetic field set for ~17 MeV tritons (nea r the energy 4 calculated for tritons from the l C(p,t) 12 c reaction corresponding to the predicted ex citation of the T = 2 state) . At this field setting, nuclear reactions on aluminum produce deuterons, tritons and alphas at the detectors, but no 3 He++ (from Q-value considerations) . The tritons and alphas were stopped in the detector while the deuterons penetrated through the active region . Thus the highest-energy events forming a narrow line in the energy spectrum were i dentified as alpha particles. E(a) 3 • The tritons were then located as a sharp line at an energy To confirm this identification, the 14 . C target was inserted, . and alpha-particle groups were observed from the 14 C(p, a ) ll . B reaction 23 corresponding to the ground and low-excited state of llB. A ·typical energy spectrum from the detectors is shown in Figure 8 . The gains were adjusted so that the alpha line was off- scale, for the triton measurements. 30. The triton group falls at ~channel The broader group at ~ channel l4 is identified as deuterons. The smooth background is produced by a high-energy tail associated with the elastic proton groups (which have nearly the same rigidity as the tritons of interest), and, in part, by neutrons. (The latter contribution was confirmed by a short run with the spectrograph field turned off) . The elastic protons apparently produce the high-energy background by scattering from nuclei within the detector along the active region; the protons thus deposit a considerably greater energy in the detector than the mean energy-loss for the protons passing through the detector. The two-dimensional data array corresponding to this energy spectrum is shown in Figure 9. The diagonal cutoff on the right side is associated with the edge of the detector (see Appendix B). triton line is again evident at ( energy) channel 30 . The The high- energy tail from the elastic protons is seen as a broad smear on the le~ side of the array. Because of the logarithmic display scheme used, the deuteron group is lost in the background in this figure . The triton spectrum was surveyed at 30 ° in the lab , over a range of energy corresponding to excitations in l to 29 .4 MeV. 2 c from 24 . 3 MeV The results of this survey are shown in a composite spectrum (Figure lO). Only one narrow g roup identified as tritons was observed above the continuum yield. The second group in the composite spectrum, to the le~ of the triton group, i s associated 24 with the elastic protons discussed above. To confirm the identi- fication of the one triton group with a level in 12 c, the 14c target was replaced by a natural carbon target of similar thickness and construction, and the tritons produced by bombarding this target were observed in the region where the group had been seen (insert in Figure 10) • The l4 C target was then reinserted and the group of inte rest was observed at 25° and 20° in the l ab. A typical triton spectrum is shown in Figure ll, for a run at 25° spanning the observed triton group. It was not seen in runs at l5° and 10°. In changing from one angle to another, the focal plane was not adjusted for changes in the kinematic broadening of the group. A calculation of this effect showed that the group should be quite broad at l0°, with the focal plane adjusted as it was for 30°. The focal plane was reset for the 10° kinematics, and a survey of the triton spectrum was again made (Figure l2). No narrow groups (other than that associated with the elastic protons) could be clearly discerned above the continuum at this angle . Because of the limited accelerator time remaining, no further investigation of the angular distribution was possible. The magnet was moved to 20° and the focal plane was suitably adjusted. The narrow group was quickly located. A careful position and field measurement was then made, with the kinematic -angl e defining slits at the spectrograph entrance reduced to improve the resolution. Several alpha-particle groups from .t he l4C(p,a)lL -~ reaction, corresponding to the ground and first-three-excited states of 1 1n, were observed for position and Q-value calibrations. 25 An alpha-particle spectrwn produced by bombarding the aluminum target was also observed at approximately the same fields, to establish clearly the edges of the detector. Some typical alpha spectra are shown in Figure 13. The 14 C target used for these measurements was brought back to Caltech 'Where the thickness and composition were measured (see Section II-C, and Figures 5 and 6). The thickness data are sum- marized in Table 3. 2. Excitation Energy For an accurate determination of the excitation energy of the observed 12 c state, the position of the group (and hence its energy) must be precisely extracted from the data. The relation- ship between pulse height, in the recorded position spectra, and position along the focal plane was constructed from a comparison of the calculated positions and observed spectra for the alphaparticle groups corresponding to states of ground~state 1 1:8. In particular, the alpha group was carefully observed at three points along the detector spanning the region where the triton group of interest had been observed. The positions of the alpha groups were calculated from the beam energy measured by the field at the 153° analyzing magnet, the measured target thickness, .the spectrograph field, and the known masses and excitations (Mattauch 1965, AjzenbergSelove 1968). A be$t-fit polynomial (in the least-squares sense) was constructed for calculating positions (Figure 14). energy for. the level in 12 The excitation c was then calculated from the position 26 determined from this calibration, the spectrogra ph field, the beam energy, target thickness and nuclear masses. The results of this calculation are summarized in Table 4, for the 12 c level and the observed alpha particle groups. The data in this table include all the observations of the triton group in the number-2 detector, chosen for the Q-value measurement. (The number-l detector was not calibrated carefully, so no final results have been taken from the data observed by this detector.) the 30° scan, the deuteron group from the 14 13 c(p,d) c reaction corresponding to the lowest T = 3I 2 state in was observed. During 13 C(E = 15.112 MeV) x The calculated e xcitation for this level is included in Table 4 for completeness. However, this observation was not used as an additional calibration b e cause: (l) a~er a small change was made in the electronic circuitry the 30° scan, but before all the observations listed in Table 4, which could have affected the position calibration, and (2) the deuterons observed fell at ~ channel 15 in the energy spectrum, while the alphas and tritons were generally observed near channel 40. Thus the zero point correction for the position spectrum is very important in comparing these spectra, and the measurement of position zero was somewhat in doubt. Energy loss in the target was calculated from the stopping power formula given by Barkas (1964), with shell corrections. The reaction. angle was not adjusted to an effective value depending on the kinematic-angle aperture (see Hensley 1969), since the focusing properties of the Oak Ridge spe ctrograph should compensate for this 27 effect in first order, and the effect is small in any case. The final value for the excitation energy was determined by averaging the results in Table 4, with the observation made at 20° with reduced kinematic-angle aperture having twice the weight of the remaining observations. Because of the procedure used for calibration which located the triton peak in relation (principally) to the ground-state alpha group, the uncertainty in the excitation energy is rather sensitive to (and dominated by) the uncertainty in the beam energy. Since the alpha-particle and triton groups were observed at the same position and the uncertainty in the position of the tritons is equal to that in the position of the alphas, the dependence of the excitation on the beam energy can be written For a uniform-field spectrograph, so the above relation beco1res dE x a.El Inserting the numerical values for the kinematic factors at 20° gives the result 28 From a comparison oi' calcul ated and obfJervctl 1'ocnl.- J11tJ.nc: positions of states with prec:lscly know Q-vaJ_ues :Ln mm1crouu nuclcJ , observed in elastic scattering or r eactions such as (p, d ) by photographic emulsions at the focal-plane, the precision of the beam energy has been found to be approximately 30 keV or less. Then the resulting excitation energy and uncertainty for this level is Ex 27 . 595 ± 0.020 MeV. 3. Angular Distribution The partial angular distribution measured for the 27.59-MeV state is shown in Figure 15. The horizontal bars indicate the spectra- graph aperture (in the center-of-mass system), while the ve rtical bars indicate the uncertainty resulting f r om counting statistics and background subtraction. An additional uncertainty of 20% i s assigned to the absolute normalization. The points indicated at 14° and 19° are upper limits estimated from the data in the region where the level was expe cted, although no group could be discerned above background at either of these angles . The target thickness used in extracting the cross section was obtained from t he measurements in Table 3 , corrected for hydrogen content in the target, assuming the polymer to be ( CH) • n Though the observed distribution is far from complete, it does not appe ar to be consistent with the usual L = 0 pattern expected for the lowes t T = 2 state, since the forward-angle peaking was not observed . However, a. DWDA cn..lculation of the L = 0 distri- 29 bution, using the potentials suggested by Cos-p=r (1967) for 10 12 c(p,t) c (Appendix c), predicts an angular distribution which is qualitatively similar to the observed distribution (Figure 15); while the detailed shape of the predicted distribution is sensitive to the radial dependence of the absorptive potential, no strong forward-angle peaking is predicted at this bombarding energy. Since the distribution measurements are incomplete and the experimental uncertai~ties were rather large, no firm L-value assignment can be made from these data. Because of the inconclusive nature of these measurements and the unorthoqox shape predicted for the angular distribution at this bombarding energy, . further study of this reaction is of considerable interest, both to extend the measurements at this energy and to measure the distribution at higher energy, where the L = 0 shape is predicted to be peaked at forward angles. 4. Width From the observed width of the triton group and the calculated resolution at the focal plane of the spectrograph, an upper limit can be placed on the width of the 27.59 MeV state. The resolution was calculated by adding quadratically the contributions from energy losses in the target, residual kinematic broadening, and broadening produced by the last beam-focusing quadrupole lens before the target chamber. The last two contributions were calculated by a computer code (available at Oak Ridge) from the reaction kinematics, s pectragraph aperture, focal plane orientation and quadrupole current. For observations of the g r ound-state alpha-particle · group and the 27.59- 30 2 MeV state of l c at . 20 °, the various contributions to the resolution and the observed widths (in particle energy) are sunnnarized in Table 5. The calculated resolution is dominated by the contribution associated with the beam focusing (quadrupole). From a comparison of the calculated resolutions and observed widths, it is apparent that this term was overestimated in the initial calculation. The quadrupole term was re-estimated by fitting the observed width of the 11 B(g.s.) alpha group, using the target and focal plane terms listed. The new value extracted for the quadrupole term was scaled by the ratio of the previous values for this term for the two groups, and this value was used to compute the revised total resolution listed in the table. This entry is consistent (at least) with the observed widths, though not reliable enough to extract a meaningful value for the width of the 12 c level of interest. An upper limit for the width can be established by dropping the quadrupole term from the resolution and calculating a minimum resolution from the target and focal-plane terms. The results for the two observations listed are r < 35 keV It is likely that this is a conservative estimate and that the actual width is considerably less than this limit. 5. Conclusions In this study of the reaction bombar~ing 14 12 c(p,t) c at 50.55 MeV energy, only one narrow triton group corresponding to a 31 state in 12 c was observed in the region of excitation energy near that predicted for the lowest T = 2 state. Because of its narrow width (for such high excitation energy), and the excellent agreement between the predicted and measured excitation energies, this level is identified as the lowest T = 2 state. Because of the ambiguity resulting from the incomplete measurement of the angu],.ar distribution, further. study of this state by the same reaction is of considerable interest, to verify the spin and parity of the state. An intensive effort has been made by Black and collaborators at the Australian National University to locate this level as a resonance in an isospin-forbidden compound-nuclear reaction. 11 B + p have ·indicated no resonances in for studies of lOB + d and the region of interest (Black 1970). the present 14 C(p,t) Results With the aid of the results of 12 C measurement, a study was undertaken by the Australian group of the capture reaction narrow range of excitation. 9 3 12 Be( He,yy) c, over a The y -decay is assumed to cascade through the lowe.st T = l state at 15.l MeV. Preliminary results of this search indicate a weak, narrow resonance corresponding to a 12 c excitation of 27.585 ± .005 MeV, in good agreement with the present experiment. The width of the resonance observed is of the order of 5 keV or less. :).. • For the used in the 14 3 1 c(p, He) Experiment 2:s study, a 14c target similar to the one 14 12 C(p,t) C measurements was employed. These two targets were prepared simultaneously; the polymer layers are expected t.o have equal thickness, within ~3C/fo, and the gold layers were expecte d to be of the same thickness within ~1Cl{o. For this run, the beam energy was 63.4 MeV. The detectors were again placed midway along the focal plane. For the initial work, a 1-mil mylar foil was placed in front of the detectors. To establish the particle identification, the energy scale in each detector was calibrated by observing the two alpha-particle groups from a ThC' source positioned directly in front of the detectors. The 3 He++ ions were then identified from combined energy and rigidity measurements. A particle group identified as alphas were 3 evident at ~ E( He) confirming the particle identification. A typical spectrum of energy deposited in the detector is sho'Wn in Figure 16. The sponding to 3 He spectrum was surveyed over a continuous r egion corre1 2:s excitations from 5 to 17 MeV, at 20° in the lab. A prominent, narrow group was observed near the predicted excitation for the 1 2:s (T = 2) state, as shown in the composite 20° spectrum (Figure 17). Another group observed at somewhat higher excitation was not as strongly populated, and in subsequent runs there was inconclusive evidence of a closely spaced doublet of levels. To confirm 4 the correspondence of the observed groups with levels in l2:s, the l C was replaced by a 12 3 C target prepared in the same way, and the He spectrum produced by bombarding this target was observed at the appropriate field settings (insert in Figure 17). The 14 c target was then r einserted and a study of the angular 33 121· A t yp1ca . 1 3H.e spect rum s h ow i ng tl ~ -ie .l . t ri'b u t'ions was b egun. d is state at 12.7 MeV excitation f r om a single run at 0° is Gltown Ju Figure 113. At J'irGt an attempt was made to ntudy tJ-ie m11-'.11lnr d :i 1: - tr:l.lmtions for lioth of the observed groups simultaneously by olnwrving one group in each detector. This was not feasible, however, since the detectors could not be placed sufficiently close together to provide unambiguous results for both groups, because of the mounting hardware used. The two groups were then observed in separate runs at 6°, 10°, and 15°, but the measurements on the weaker group (or doublet) at higher excitation had to be discontinued to save time. The lower level was then observed at 8° , 12.5°, 17.5°, 25° , 30° , 35°, and 41° to provide a more complete angular distribution. For these measure- ments, the focal plane was readjusted to·the proper position for each · angle. With the spectrograph positioned at 8° in the lab and a 5-mil Al foil added in front of each detector (to provide better particle identification at higher particle energies), careful position and field measurements were made for the 12.7-MeV state, for a Q-value determination. of 12-- Then the 3 He group corresponding to the ground state ' l3 was observed for a Q-value calibration. Finally, the 5-mil Al foils were replaced by ladder masks of 1 0-mil Al. Alpha particles from a ThC' alpha source were observed through these masks, to provide zero determinations (see Appendix B) and position-dispersion measurements. The dimensions of the masks were measured later by a travelling microscope. '.l'Ilu <::xcJtatJon lJ ner.-g:Lt~IJ l.'ut• tile· ·two J.< ?VL!J 11 ul>uerv•?d .I 11 l.111• region of interest were calculated from the e xiting particle energy measured by the spectrograph, the beam energy determined by the beam-analyzing magnet, the target thickness measured for the matching 14 . C target (Table 3), and nuclear mass tables (Mattauch 1965). For the spectrograph energy measurements, a position calibration curve was constructed, for one of the detectors, from the ThC' alpha-particle spectra observed through the ladder mask (Figure 19). The effective center of the detector was adjusted to give zero ex1 citation energy for the ~-ground-state group observed at 8° . The center determined in this way agreed within 0.5 mm with the nominal center of the detector. The second detector was not calibrated precise ly for Q-value calculations, since most of the measurements were made with the f i rst detector, including all the observations of the calibration groups (principally the 1 ~ ground-state group) . The results of these calculations are presented in Table 6 for observations of the 12.7-MeV state at seven angles , and for one observation of the 14.9-MeV state. calculated for the 1 In addition, excitations ~ ground and first-excited states and for the first excited state of 1 4N (observed at 8°) are included. For the determination of the final value for the excitation energies, the values in Table 6 were averaged, with the results for 6° and 8° having twice the weight of those from the other angles because of the higher resolution used for these measurements, and 35 the higher yield. The uncertainty in the result was calculated by adding quadratically the contributions from the uncertainty in beam energy (see Section IV-B-2) and that from the uncertainty in exiting particle energy resulting from uncertainty in position, estimated from the scatter in Table 6 . The final result for the excitation energy for the 12.7-MeV level is E = 12.710 ± 0.020 MeV and for the 14.9-MeV level is E 14. 860 ± 0.030 MeV. 3. Angular Distribution The angular distribution measured for the 12.7-MeV state and the pQrtial distribution measured for the 14.9-MeV state are shown in Figure 20. The horizontal bars indicate the spectrograph aperture (in ·the center-of-mass) and the vertical bars indicate the uncertainty resulting from counting statistics and background subtraction. An additional uncertainty of 2C/fo is assigned to the absolute normalization. The target thickness used to extract these cross sections was taken from the results in Table 3, corrected for hydrogen content, assuming the polymer to be (CH) • n The smooth curve shown for the angular distribution of the 1 2 . 7-MeV state is the calculated L = 0 distribution from the code JULIE. The details of' the potentials used and the method oJ' cul - culation are summarized in Appendix C. The experimental distri - bution is clearly in good agreement with the predictions for L = O. The partial distribution for the 14.9-MeV group is insufficent to make an unambiguous assignment of L-value. 4. Width From the widths of the observed 3 He groups and the calculated resoJ_ution at the focal plane of the spectrograph, estimates for the widths of the two observed levels have been made. For several observations at 8° with a kinematic aperture of 2° .and at 10° with a 4° aperture, the contributions to the resolution for the ground state and two excited states of interest, and the observed widths, are summarized in Table 7. For these measurements, the calculated resolution is dominated by the energy loss in the target. The estimated energy-loss strag - gling in the target is small compared with the r emaining terms , and has been neglected. The observed width of the ground-state group is somewhat larger than the calculated resolution. This may result in part from the position resolution of the detector itself, since the group is ~uite narrow in distance along the focal plane (~l.9 mm FWHM). By assuming that the difference in observed and calculated width for the ground-state group arises from this source (only), an additional term can be extracted for the resolution function, corresponding to a detector position resolution of l.5 mm (FWHM). The corrected resolution for the 12.7-MeV state determined in this 37 way is listed in the last column of' Table 7. Using this calculate d resolution, the width extracted f'or the 1 2 . 7-MeV state j _s r(l2.71) = 27.5 ± 1 8 keV where a 15 keV uncertainty has been assigned to the corrected resolution. An upper limit for this width, obtained from the un- corre cted resolution listed in Table 7, is r(12.11) < 54 Kev Since the width of the 14.9-MeV group is considerably larger than the calculated resolution, it can be extracted with more confidence, if it is assumed to be a single state. Using the cal- culated resolution, the width is found to be r(14. 86) (226 ± 30) keV • A lower limit f or this width can be found by assuming that the width of the 12.71 MeV state measures the resolution, and by adjusting this width for the small difference in energy loss for the two groups. The result of this calculation gives r(l4 . 86 ) > 221 keV • 5. In this study of' the Conclusions 14 3 1 c (p, He) 2:s reaction at 63 .44 MeV bombarding energy, two reasonably narrow groups corresponding to states of 1 2.s were observed in the region of excitation energy between lO and 15 MeV. Because o:r the agreement l>etweeu the pre- dieted and measured excitation energy, the small width, and the L = 0 angular distribution for the 12.7l-MeV state, this level is identified as the lowest T = 2 state in l2_s. There is insufficient evidence for an isospin assignment for the l4. 86-MeV state; the width is somewhat larger than would normally be expected for a T = 2 state unless the observed group is, in fact, a doublet of [This level is unbound by only ~300 keV to the isospin- levels. allowed channel lOB + d*(T = l)]. D. Prediction of the Mass of l2:se Since the report of the discovery of the particle-stable isotope 1 2.se (Poskanzer 1965 ), no measurement has been made to accurately determine the mass of this nuclide. An estimate for the mass can be determined from the estimated beta-decay endpoint energy, in the reference above. Cerny (1968) has quoted an upper limit for this mass, presumably based on the estimated upper limit for the beta-decay endpoint energy. A prediction by means of the Garvey- Kelson mass formula (Garvey 1966) is lower than this upper limit by 3.3 Mey . It is of interest to use the measured excitation energies of the lowest T = 2 states in 12 c and 1 2:s to predict the mass of 1 2:se, for a comparison with these values. From the quadratic nature of the mass law, it is readily apparent that a prediction of the mass o:f 1 2ne cannot be mo.de dir- ectly :from the excitations o:f the two T = 2 s tates alone, since there are three constants to be determined. Two schemes can be proposed to provide an additional number to use f'or determining the constants: (1) Assume the T-dependence of the coefficient of T z can be neglected, and determine this coefficient from the mass difference 1212-_ . 6(1) of the T :::; 1 ground-state masses ( -.S - ~). . (2) Use the Garvey-Kelson Formula for predicting the (T :::; 2) 12 1 mass difference 6(2 ) :::; ( 2:Be 0 ), from which the coefficient of T z can be determined directly. There are indeed a number of mass relations attributed to Garvey and Kelson (e.g., see Garvey 1969) . In the latter procedure outlined . above, two such relations can be used, based on T :::; 1 or T :::; 1/2 nuclei, respectively: and where the nuclide symbols represent the ground-state mass of the nuclide indicated. From the point of view of (1) above, the difference 6(2) could be written 6(2) 26(1) The G-K relations given above are thus seen as the use of a Tz-coefficient for T:::; 2 nuclei, obtained by averaging the Tz coefficient for T :::; 1 or T :::; ~ nuclei over mnss number in the region of interest. In a review article on Coulomb-energy s yuternnt:i cu , Jl:l.ne cke (1969) has concluded that the vector Coulomb energy (i. e ., the coefficient of T ) to be essentially independent of T, for z T = 1/2, T = 1, and T = 3/2 data over a wide range of nuclei, and to depend approximately linearly on A within a given subshell. Then to the extent that the T-independence of the vector Coulomb energy is also valid for T = 2 nuclei, either of the schemes proposed should provide an accurate estimate for the mass of 1 ~e . For either scheme, the resulting mass prediction b e comes l~e where the nuclidic symbols now represent the T = 2 excesses of the nuclide indicated. dictions for 1 The ~uantities 6~T) and the resulting mass pre- ~e are summarized in Table B, along with the mass estimate and the uppe r limit established from the beta-decay endpoint energy and the G-K prediction. 41 V. 2 T == 2 STATES IN A. °F Introduction The identification of the iowest T == 2 states in 2 °F and 20 Ne was first reported by Cerny (1964) from studies of the re. 22_ 20 22 .. 3 20 actions ~e(p,t) Ne and ~e(p, He) F. These measurements have been repeated to provide more precise Q-value determinations (Hardy 1969). The precision of the excitation energy for the lowest T == 2 state in 2 °F, from these more recent measurements, is 35 keV . The T == 2 analogue state in 20 Ne has been located with a precision of 6 keV by Adelberger (1967a,b) in a study of the reacti·on 18 3 o( He,n) 20 Ne. The res ult . s o f these measurement s h ave b een confirmed and improved on by the observation of this state as an isospin-forbidden compound-nuclear resonance (Block 1967, Kuan 1967). The precision of these combined measurements of the excitation energy of the lowest T == 2 state in 20 Ne is 2.8 keV. The excitation energy for the lowest T == 2 state in be predicted either from the known mass of of the analogue state in 20 20 F can 0 or from the excitation 20 Ne, according to the relations given in Chapter III. These predictions are summarized in Table 9, together 20 . with the results of the F measurements discussed above . . The present s t u d y o f the react ion 18 3 o( He·, p) 2 0F '·""'S w= und er- taken shortly a~er the work by Adelberger (discussed above) on the 2 20 Ne analogue state, to provide a value for the °F(T == 2 ) excitation energy of comparable precision. The details of these measurements are discussed in the following section. Section C deals with the 42 determination of the excitation energies for the two states of interest. The angular distributions for the proton groups corre- sponding to these states are discussed in Section D. The last section summarizes the results and conclusions from this study. B. Experiment 18 3 2 All the measurements of the reaction o( He,p) °F under consideration in this chapter were made at an incident energy of 12 MeV. The first data obtained for this study were collected simultaneously With deuteron and alpha spectra, by the use of the sixteen-detector array at the focal plane of the Caltech spectra. graph . The results for the study of l8 3 0( He,a) l 7 O have been re- ported separately (Hensley 1969); since this study was of considerable current interest, the measurements of the proton spectra were not optimized at that time. In particular, singly-charged 3He ions were not adequately separated from the protons (which were passing through the detectors), so that a considerable background was superimposed on the proton spectra. Measurements were made both 18 with self-supporting Ni o targets and with a gas - cell filled wit h l8 0 gas. The gas -cell target had a considerably smaller fraction of impurities (particularly 12 c and 160) than the Ni18o foils, but the experimental resolution was also considerably poorer. Because the proton spectra contained many groups in the region of interest, the gas-cell measurements were not adequate to allow separation and identification of the groups arising from 18 0, 16 0 and 12 c. 18 The final data were taken with a freshly made Ni o target (to keep the l.::'c and J_GO conti:unj_nn.tj_on no mnn.lJ_ u.o po:.;t;Jl1lc), und a thick foil was placed over the detector array so that on.ly protonc were observed. Although the loss of energy calibration groups from the deuteron and alpha spectrum was regrettable, the improvement in background mdre than justified this procedure. Pr oton spectra were collected at 10° and 20° with this arrangement. The 10° spectrum is shown in Figure 21, along with a similar gas-cell spectrum and contaminant spectra ( 16 o and 12c). A prominent group is apparent at the expected position of the lowest T = 2 state in the spectrum, corresponding to a state at 6.51-MeV excitation in 2 °F. At ~l. 7-MeV higher excitation a weaker group was located. This group was seen more clearly in the 20° spectrum (Figure 22). With a thin Ni 18 o target, to provide good resolution of the groups near the 6.51-MeV state,the angular distributions was measured in the position-sensitive detector. A typical position spectrum is shown in Figure 23. Because of the difficulty in separating the 8 .12-MeV state from an adj a cent proton group, only incomplete distribution measurements were ma de f or this group of protons. C. Exci tation Energy The excitation ene r gies of the two state s o f interest were extracted from t he proton s pectra measured at 1 0° and 20° in the sixteen-counte r array. (f rom The 18 F ground-state group f r om 16 3 18 0( He,p) F 1 6 0 . . •t• . the Ni18 o target) was chos en for calibration, impuri i e s in s ince it falls ne a r the 6 . 51-MeV 2 ° F group i n t he p r oton spect rum . Since the gr oups of i nte r est a r e a ll protons of comparabl e energi es , the Q-value me aourernents are relat:lve ly J.nLw11:.Li.t :Lvc~ ·Lu t~rrur :; .l 11 'LJ1e spectrometer calibration, or to an incorrect (but consistent ) assumption about the shape of the peaks in dete nnining the central frequency. 16 If the distributions of 0 and 18 0 in the target are 18 similar, the energy losses are comparable for the F calibration . 20 . group and the 6 .51-MeV F group, and consequently the excitation energy extracted from these data is insensitive to the target thickness. The distributions of the oxygen isotopes in this kind of target has been studied in some . detail by Hensley (1969 ), in his study of the 16 18 o( 3He,a) 17o reaction. He found no evidence that the 0 distribution is unsymmetric, with respect to the center of the target, and no disagreement between calibration groups based on 1 6 0 and 18 0 i n the target. The 18 0 distribution is expected to b e synn:netric from the procedure used to produce the targets. the present analysis the distributions of 16 0 and 18 Thus for 0 in the target were assumed to be symmetric, and the same nickel and oxygen thicknesses were used for calculating energy losses for all the proton groups under consideration. Since deuteron and alpha spectra were not collect~d simultaneously with the proton spectra, no additional calibrations could be obtained from these particies. groups from the reaction 12 1 c(He,p) The proton 4N were not used for calibration purposes, because of the uncertainty in the location of the 12 c impurity within or on the targets. The excitation energies extracted from the data are presented in Table 10, for the two observations of each of the two of interest and the 18 F calibration group. 2 °F groups The tabulated uncertainty 45 :i.B only that contribution :front the est:l.m ated uncerta:l.rr1.:y :11t lu cnt:t.np; the centro:l.d. o f cu.ch group . J\.d.dit:lonn.l ernrLr:tlmL:t01111 l:1i tl1<? 111H:r!l'- tainty of the final results arise from: 16 18 . (1) the uncertainty in the O(He,p) F Q-value (2) the uncertainty in the calibration constant of the spectrometer (3) the uncertainty in target thickness. These contributions are listed for the two 2 °F states of interest in the table below. Contributions to the Uncertainties of 2 °F Excitation Energies (all e ntries in keV) E x Q-value Spectrometer Target Frequency Total 6513 · ..... , .4 .2 1.6 2.8 8210 1 .6 .3 2. 4 3.4 To allow for possible small differences in the 160 and 1 8 0 distrib utions in the target, an additional (arbitrarily chosen) contribution of 2 keV was folded in t o obtain the total uncertainty given in the tabl e . The final results for the Q-value measurements are shown in the table below. Since the ground-state mass excess of 20 F has been assigned an uncertainty of 4 . 7 keV (Mattauch 1965), the reaction Q-values as well as the excitation ener gi es are presented . The un- certainties given for the excita.t:Lon e nergie s include the ground- 46 state mass uncertainty. Q-values and Excitation Energies for T = 2 States in 20F State Q-value (keV) Excitation (keV) Lowest T = 2 +258.5 ± 2 . 8 6513 . 3 ± 5 . 5 First-excited T = 2 -1338.5 ± 3.4 8210.3 ± 5 . 8 D. .Angular Distribution The angular distribution measured for the 6 . 51-MeV state is shown in Figure 24. The experimental errors shown are the result of counting statistics and a 5% uncertainty in the beam charge integration. An additional uncertainty of 3cy/o is assigned to the absolute scale to include the target -thickness uncertainty for the thin Ni 18 o target used for these measurements. The experimental points at 50° and 60° are estimated upper limits; the group was not identified above background at these angles. The smooth curve shown in this figure is an L = 0 DWBA calculation for this reaction. The optical potentials used for this calculation are discussed in Appendix c. The excellent agreement between the measured and calculated distributions allows an unambiguous assignment of L = 0 for this transition. A complete angular distribution was not obtained for the 8 .21 MeV state because of the difficulty in separating this group from nearby groups . From the observations at 10° and 20°, this 47 group appeared to be the only one present, consistent with L = 2, · near the predicted excitation of the first-excited T = 2 state. I E. Conciusions I ! The T = 2 assignment for the 6.51-MeV level is confirmed by the L = 0 angular distribution and the good agreement of the measured excitation energy with that predicted for the lowest T = 2 state. The excitation energy determined in this experiment is in agreement with the results from the 2 ~e(p, 3He) 2°F measurements of Hardy (1969). A T = 2 assignment for the 8.21-MeV state is tentative, based only on the difference in excitation energy from that of the 6 .51-MeV state, and the consistency with an L = 2 assignment from the observations at 10° and 20°. assignment. More complete data are required for an L-value The measured excitation energy is roughly consistent with the early results presented by Cerny (1964) but falls slightly outside the probable error quoted for that measurement . 48 VI. T 2 STATES IN A. Introduction 28 Al In a tabulation of the known T = 2 states in A = 4n, T = 0 or 1 nuclei, Cerny (1968) listed excitations for all such nuclei 28 28 for 12 < A < 40 with the exception of Al and si. In a letter published later that year, McGrath (1968) presented data for the 28 28 30 reaction si(p,t) si from which the lowest T = 2 state in si was identified. Along with the excitation energy extracted for that state, a result is quoted for the excitation of the lowest T = 2 state in 30 28 28 3 Al from a study of si(p, He) Al . The data and final results from this work were presented in a more recent publication by the Berkeley group (Hardy 1970). An independent study of the 28 Al T = 2 state, in the reaction 3 28 26 Mg( He,p) Al, has been reported by Clark (1970). In this work, the T = 2 state was identified from the predicted excitation energy and angular distribution, and from a comparison of the for various levels in the reactions average yields 27 26 3 28 28 Mg( He,p) Al and Al(d,p) Al. The usefulness of such a comparison -was considered briefly in Chapter III; although the isospin conservation should inhibit the T = 2 state in the ( d,p) reaction, the effects of nuclear structure should be conside red carefully, since the two reactions being compared are rather dissimilar. At best , the comparison can provide only a negative confirmation (a level strongly populate d in (d,p) is not likely to b e T = 2 ) or consistency check, as in the present case . An additional difficulty ari ses in this particular case from 49 the high density of T = 1 levels in the region near the T = 2 state in 28 Al; very precise measurements are required to confirm that the same levels are indeed being considered in the two reactions . In the study in question, no discussion of the precision is given; the excitation energy was taken from a compilation of levels of 28 Al (Endt 196 7) • Table 11 sunnnarizes the predicted excitation of the lowest T = 2 state in 28Al and the results of the previous measurements of this excitation from the experiments discussed above. The study of the reaction 3 26 28 Mg( He,p) Al was begun at Caltech before the publications described above had appeared, and the work was continued to complement the work already completed here and in other laboratories. Since the Caltech program has con- centrated on precise Q-value measurements, and the pre cision of the work by Clark is uncertain, there is no serious duplication of effort. In addition to the study of the lowest T = 2 state, special attention has been given to the excitation energy region near that x ~ 7.5 MeV ). predi cted for the first-excited T = 2 s tate (E The next section describes the details of the experimental measurements. Section C deals with the determination of the excitation energies for the states observed. The angular distributions of the p r oton groups are discussed in Section D. The results and conclusions from this study are summarized in the final section. B. Experiment All measurements of the reaction 26 3 28 Mg( He,p) Al were made at 50 l2-MeV incident energy. Because of special difficulties in pro26 ducing thin transmission targets of Mg from 26 . MgO (see Section 11-ll) the initial meacurements were made on a to.rgct w:Lth carl)on backing. the predicted 10014:~/ em~~· From a comparison of' Q-values, it is ev:Ldent that 28 .Al (lowest T = 2) group should be nearly coincident in energy with the proton group corresponding to the first-excited state of 14-_ ~. ( 14-_ ) 12 3 )14__ . N , from the C( He,p ~ reaction. 1 first measurements, the from 1 In these ~ groups were much stronger than those 28 .Al; ho"Wever, a definite shoulder was observed on the high1 energy side. of the resolved group. ~1 group indicating the presence of an un- Subsequently, 16 Mg targets were successfully produced on thin gold backings, and new data were collected. A comparison of proton spectra is shown in Figure 25, for the goldbacked 26 Mg targets, and targets of the principal contaminants expected in the 26 Mg target ( 12 c, 160 and 2 ~). plotted from observations made at 10°. observed in the spectrum from the These spectra are A strong proton group is 26 Mg target close to the predicted excitation of the lowest T = 2 state. A comparison of the 1 4N ground-state group in this spectrum with that in the spectrum from a carbon target confi rms that the group of interest was not produced by 12 c impurities in the target. The sixteen-detector array was used for precise Q- value measurements; proton, deuteron, and alpha-particle spectra from induced reactions were collected at several angles . He- Considerable care was tal1:en to separate the protons from s ingly- charged particles, in these measurements. 3 3 He The target t hickness was men.sured 51 by the elastic~scattering techniques described in Chapter II. The angular distributions of several proton groups near the lowest T = 2 state were obtained during the above measurements. For the final angular distribution measurements for the lowest 'I' = 2 state, a complete distribution was measured using the positionsensitive detector at the spectrograph focal plane. A typical position spectrum is shown in Figure 28 . The array data were studied caref'ully in an attempt to locate the first-excited T = 2 state. At forward angles, the region of interest contained proton groups from the first several excited state of 18 F. From a comparison of the proton spectra from 16 0 and 26 26 Mg in Figure 26, it is evident that the oxygen content of the Mg target is reasonably small, as shown by the 20 F ground-state group. By a caref'ul consideration of relative yields in these two spectra, the proton group at 33.35 MHz has been identified as a state in 28Al. A detailed spectrum from this energy region is shown in Figure 27, for 20°, where the composite group is partially resolved into a doublet . The 28 Al identification is substantiated from kinematic tra cking at 20°, 30° and 40° . C. Excitation Ene rgy The primary Q- value calibration was taken from the deuteron 27 group corresponding to the lowest T = 3/2 state in Al at ( 6 . 815 ± . 002) MeV (Endt 1967). obtained from the Additional calibration checks were 25 Mg - gr ound-state alpha group and the tritons . f eeding the 1.059 -MeV state of 26Al ( Endt 1967 ) • The rigidity of 52 2B the deuterons was quite similar to that of' the ( · Al) proton group of' interest; the al.phas and tri tons had higher r:igid.:Lty, oirn:LJ..nr to that of the grou.nd-otatc protonu. Both the deuteron grou.p nrnl the alpha group were strongly populated at forward angles; the triton group was considerably weaker. The ground-state proton group was also weak and could not be resolved from the first-excited 28 Al T = 2 proton group was very state (E = 31 keV). Although the x 18 .. 16 close to the · F ground-state group (from 0 contamination in the target) and the 1 4N first-excited state (from 12c in the target), these groups were not used for calibration because of the uncertainty in the location of the carbon and oxygen in the target, relative to the magnesium. For the Q-value cal culations the ground-state masses were taken from recent tabulations (Mattauch 1965). Excitation energies calculated from the data are presented in Table 1 2, for several observations of the 6 . 00-'MeV state and the members of the 7. 5-MeV doublet. The excitations calculated for the calibration groups dis~ussed above are also included in this table. The uncertainties listed in the table ·are the contri- butions to the total uncertainty associated with the determination of the central frequency for the group under consideration. Addi- tional contributions to the probable error of the Q-value arise from uncertainties in the target thickness, spectrometer calibration and calibration Q-values. These contributions and the resulting assigned probable error of the Q-values are listed in the table below. 53 Contributions to the Uncerta:inties o:f Measured Q-VaJ_ueu (all. entr:i.eo :l.n JwV) Frequency Q-value Spectrometer Target Total Lowest T = 2 2.5 2.7 1.5 1 4.1 7.5-MeV doublet 10 2.7 1.5 l 10.5 State The reaction Q-values and excitation energies determined for the three 28 Al states of interest are presented in the table below. The uncertainty of the excitation energy in each case was determined by combining the uncertainty for the. corresponding Q-value with the uncertainties of the ground-state masses of 26 . Mg and 28 Al (1.8 keV and 3. 7 keV, respectively). Q-values and Excitation Energies for States in 28 .Al Q;..value (keV) Excitatioµ (keV) 2287.0 ± 4.1 5996.6 ± 5.8 835 ± 11 7448 ± 11 792 ± 11 7491 ± 11 D. Angular Distribution The ·angular distribution for the 6.00-MeV state is shown in Figure 28 (from the measurements made with the position-sensitive detector). The experimental unce rtaintie s are the result of counting 54 statistics and background subtractions . An additional uncertainty of lO% is assigned to the absolute nonnalization, o.risJne; from uncertainties in target thickness and beam integration. The smooth curve shown is an L = 0 DWBA calculation for this reaction. The optical potentials used for this calculation are discussed in Appendix B. The good agreement between the calculated and measured distributions allows an unambiguous assigmnent of L = O for this transition. The doublet observed at ~7.5 MeV excitation was studied carefully in the array spectra, in an attempt to extract angular distributions for the separate members. The distributions extracted for these two groups are shown in Figure 29. As a result of the dif- ficulties in separating the two weakly populated groups in the region where the background is somewhat uncertain, the errors are rather large. The points indicated at 32° and 42° for the 7.45-MeV state are upper limits only; the group was not clearly resolved from contaminant peaks at these angles. Calculated distributions for L = 1 and 2 are shown with the measured distributions. The distribution of the 7.49-MeV state is in reasonable agreement with the L = 2 distribution; that of the 7.45-MeV state is more likely to be L = 1, though no firm assigmnent can be made. E. Conclusions From the L = 0 angular distribution and the agreement of the measured excitation energy with that predicted for the lowest T = 2 state, the 6 .00-MeV state is identified as the lowest T = 2 55 28 Al. state in From this T = 2 assigmnent, the spin and parity for this level are inferred to be o+. From an examination of the doublet near 7.5-MeV excitation, the member at higher excitation is tentatively identified as the first-excited T = 2 state, on the basis of the L = 2 arigular distribution and the agreement of the measured excitation (relative to the 6.00-MeV state) with the known level 28 Mg. spacing of A corresponding tentative assigmnent of spin and parity 2+ is therefore made for this state. Although the measured excitation energy of tti.e 6.00-MeV state is in reasonable agreement with that of a state identified in the 27 AJ.(d,p) 28 Al reaction (5989 ± 10 keV) (Endt 1967), there is in- sufficient evidence for assuming these to be the same state. In particular, if such an identification is made, it is of considerable interest -- not that the level is somewhat inhibited in the 27 Al(d,p) 28 .AJ. reaction, as Clark (1970) has pointed out -- but rather that it is so strongly excited in an isospin-forbidden reaction. This . would imply an unusually large admixture of T = l strength in the state. While it seems likely that two different states at nearly the same excitation have been observed in the two reactions, it would be worthwhile to attempt to confirm this hypothesis by additional highresolution studies of the two reactions. 56 VII. T = 2 MULTIPLETS AND THE QUADRl\TIC MASO EQUATION A. Introduction In recent studies of T = 3/2 multiplets, the quadratic mass law has been tested by a comparison of measured masses for the members of completed quartets to the predictions of the mass formula. Although six multiplets have been completed, for A= 7, 9, 13, 17, 21, and 37 (JHnecke 1969), only one (A= 9) has precision mass measurements for all four members. In this case, as well as the other five listed, the deviations of the measurements from the predictions of the quadratic mass formula are relatively small and appear to be consistent with the contributions expected from higherorder perturbation terms (Nettles 1969). The T = 2~ A = 4n multiplets discussed in this report cannot be used at present to test the quadratic nature of the mass law, since mass measurements have been made for on1y three members of each multiplet (except A = 8 and 12, where only the masses for two members have been measured). However, if the quadratic law is assumed to be exact, the three coefficients for this formula are determined by the three measured masses for each multiplet. In his review article, JMnecke (1969) has investigated the 2 (A,T) dependence of Coulomb energies E~l) and EC( ) (determined from the coefficients of the mass equation by the relations given in Appendix D) for T = 1 and T = 3/2 multiplets. It is of some interest, then, to compare the results obtained from T = 2 multiplets with those 57 previous r esults. In the following sections , the quadr atic f orm of the maso formula :is assumed to 11e exact. In Section B, the Cot1loml1 c ncre;:ieu a.nd the isoscalar term determined for T = 2 multiplets in A = e, 1 2, 1 6 , 20, 24, and 28 are discussed. Section C deals with some of the predicted properties of the unobserved members of these multiplets. B. Mass-Equation Systematics for T = 2 Multiplets The measured masses for A = 4n, T = 2 multiplets are sununarized in Table 14, with references. The simple formulas used for calculating the coefficients a, b, and c f or the mass law 2 m(A,T,T) = a (A,T) + b(A,T)Tz + c(A,T)Tz are given in Appendix D, along with the definition of Coulomb energy t erms. Table 15 sunnnarizes the coefficients a, b , and c determined for the A= 4n multiplets. For A= 8 and 12, t he assumption out - lined in Chapte r IV that b is independent of T has been u s ed to extract the c coefficient. The Coulombenergies E~l ) and E~ 2 ) are also tabulated. In Figure 30, the vector Coulomb energies E~l) for T = 2 multipl e ts are compared with those extracted from T = 1 multiplets . The experimental u ncertainties are l ess t han 20 keV i n all cases . No T = 2 data are given for the p-shell multiplets (A = 8 and 1 2 ) since masses for only two members of the multiplets are known. A definite breal~ in the curve 1.s seen at A = 1 6, corr esponding 58 to the closing of the p-shell. The T = l data at A = 16 are in question, since there is an inversion of the analogue levels in 16 0, resulting from substantial Thomas-Ehrman shifts, since the T = l states are unbound in 16 It is nevertheless in- 0 and for mass-16 is much closer teresting to note that the value of to the extrapolated s-d shell results than to that of the p shell, for both the T = l and T = 2 multiplets. If the Coulomb energy is approximated by that for a charged sphere of radius r A1 / 3 , then 0 the vector term E(l) should be proportional to (A-l)/Ji2-/ 3 • From c the data, it appears that the A-dependence is more nearly linear within a subshell, as already noted by Jl!necke. The vector term appears to be nearly independent of T for the T = l and T = 2 multiplets in the s-d shell, which supports the validity of the procedure used in Chapter rv for the 1 2:Be mass prediction. 2 The data for E( ) for T = l and T = 2 multiplets are come pared in Figure 31 . The experimental uncertainties are less than 10 keV for all cases except the T = 2, A = 8 and A = 12 data, where an uncertainty of 30 keV is assigned to take account of the procedure used to extract these numbers. There is an additional uncertainty of 1 0 keV for the A = 8 value from experiment. As noted by Jlinecke, the T = l data are divided into two classes, for A = 4n and A = 4n + 2 respectively. With the exception of A = 12, there is rough agreement of the T = l and T 2 values for A = 4n nuclei. The effects ob - served for the T = l multiplets at A = 16 and 28 may be the result either of shell closure or of large energy shifts resulti ng from threshold ei'i'e cts or isospin mixing. li'rom the charged-sphere approximation, I\ -J/:·i. the 1\-depcrnJ.cncc of' the tcnoor tenn io predJ.eted to l>e 1°' r'OTll the available .data, there is insufficient evidence to t ent thin prediction decisively; the data (for T = 2 multiplets, except A = 1 2 ) are consistent with an E(c 2 ) that is nearly independent of A. . In Figure 32, the isoscalar coefficients, a, obtained from the T = 2 data are compared with those for the lowest T and the T = 0 ground-states for A = 4n nuclei. l multiplets These values, which are dominated by the binding energy arising from the strong nuclear interactions, show definite evidence for the shell closure at mass 16, in the T = 0 and T = l values, but no comparable effect for the T = 2 data. C. Predictions for Unobserved Proton-Rich Members of T = 2 Multiplets The relationship predicted from the q_uadratic mass law for the masses of Tz = -l or -2 members of a T = 2 .multiplet, in terms of the T = O, l, and 2 masses, is given in Appendix D. z thi s relation, the masses of the unobserved T z Using = -2 nuclei 120, 16 20 . 24 . 28 . Ne, Mg, Si, and S were calculated from the data in Table 14. These masses are compared with the p and 2p thresholds in Table 16. Fram this comparison, it is evident that the nuclei 20 Mg, 24s i and 28 s are expected to be stable with respe ct to h e avy-particle decays 16 12 while 0 and Ne should be unbound. The masses and excitation predicted for the T = 2 analogue in the corresponding Tz = -1 nuclei are SUllill1.arized in Table 17 and compared with the isospin-allowed de cay channels, p + (A - 1, T = 3/ 2), and 2p + (A - 2, T = l). The GO levels in 24 20 28 Na, .AJ.. and p are thus expected to be bound relative to isospin-allowed decays, while those in 1 ~ and 16F arc unbound to isospin-allowed channels. D. Conclusions The data for T = 2 multiplets in A = 4n nuclei are not yet complete enough to test the accuracy of the quadratic mass formula for isobaric multiplets. However, these data do provide values for the mass formula coefficients, under the assumption that the formula is exact. The . comparison of values of E(l) (related to the coefficient c of T ) for T = l and T = 2 multiplets supports the assumption often z made that this term. is practically independent of T. The dependence on mass number is very nearly linear within a subshell, with a distinct break at the shell closure. The tensor Coulomb energy term 2 E( ) (related to the coefficient of e T2z ) shows no dist inctive de - pendence on A or T, though the values for T = 1 apd T ~ 2 are somewhat different. It is likely, since this term is most sensitive to the omission of higher-order terms in the mass formula and to energy shi~s resulting from isospin mixing or threshold effects in the members of a .multiplet, that systematic effects for. E( 2 ) cannot be c firmly established until additional members of these mu:J_tiplets are observed, to establish in each case whether the deviations from the quadratic formula are small . The isoscalar coefficient a, which is dominated by the nuclear binding energy terms, reflects the energy required to promote particles from the lowest allowed configuration 6l (T = O) to the f'irst configuration in Which the particles can be coupled to T = 2. This term is very nearly linear in A in the s-d shell multiplets; the relatively large variations at A = 8 and l 2 are not understood. From the assumed quadratic mass f'ormula, the masses f'or the proton-rich members of the A = 4n multiplets f'or A = l2 through 28 have been predicted and their stability to heavy-particle decay estimated. It is of' considerable interest in the extension of these studies to attempt to measure the masses f'or these nuclides, and to observe the analogue state s in the T z = -1 nuclei. . 62 VIII. CONCLUSION 3 3 The two-nucleon-transfer reaction ( He,p), (p, He) and (p,t) on T = 1 target nuclei have been utilized quite successf'ully to excite T 2 states in T = 0 or 1, A = 4n nuclei. Of these three reactions, considerable experience has been developed in this labo3 ratory only with the ( He,p) reaction, at Van de Graaff energies. This reaction has become less selective in populating T > ITzl states as the studies have extended beyond the p-shell. In partic- ular, the relative strength for populating excited T = 2 states is considerably weaker at A = 20 or 28 than for (say) A = 16 (Hensley 1968). This results in part from the fact that the T > IT z I states I of interest have occurred at successively lower excitation energy as the studi es have been extended up in mass number, so that the adjacent T< states have been relatively simpler in structure and hence have competed more strongly in the reactions observed. The use of magnetic analysis techniques has been extremely successf'ul and rewarding, in these studies, for clarifying particleidentification and facilitating precision Q-value measurements. The 3 3 3 observations of reactions ( He,d), ( He,a) and ( He,p) simultaneously bavebeen useful for providing Q-value calibrations for the observations of interest, as well as additional data on states with T > IT z I · in other nuclei. The measurement of masses for several members of a given multiplet has made possible the prediction of the masses of the remaining proton-rich members and the avo.:1.la111.e deco.y modes for these 63 nuclides. In the continuation of these studies, it will be of' great interest to explore new reactions which might produce the stn.llle T = 2 , proton-rich nuclides and allow a mass measurement. is of great interest to obtain a measurement of the that of the T = 2 state in shell. 8 In addition, it 1 2se mass and Li, to complete these studies in the p- The latter measurements should be possible in the near future by means of two-nucleon transfer reactions on the high-quality lOBe targets now in preparation at Brookhaven and Los Alamos. Other more esoteric reactions have been proposed, such as ( 7Li,2p) or ( 7Li, 2n) but these offer considerable experimental problems and are not so likely to be favored by nuclear structure considerations. The extension of the study of T = 2 state s can also be attempted in the A = 4n + 2 nuclei, by using charge-exchange reactions 3 3 7 7 such as (p,ri), ( He,t), (t, He), ( Li, Be), ( 7Li~t), etc. The T = 2 states are less likely to be bound in these cases (since the parent nuclei are odd-odd). These studies are expected to be more difficult, too, because of the uncertainty in reaction mechanisms for react ions such as those outlined above, and as a result of the uncertainty in predicting spins for the parent nucle.i. The uncertainty in interpreting the angular distribution for the observed lowest T = 2 state in studied further. 12 c requires that this state be In the systematics discussed in the previous chapter, there were noticeable deviations for A = 12 which may be associated with this difficulty. It will be very interesting to compare the present results with those from reaction becomes feasible. 10 12 3 Be( He,n) c studies, if the latter G'1 Jt'inal1.y, the extenoion oi' thece utw1:Le1J ·Lo tlte 111(~nLll.ll'c111cnt 01' isospin-forbidden decay channels, such as the recent studies o:i' Adelberger and McDonald in T = 3/2 multiplets (e.g., Adelberger l969b), and McGrath (1969) in T = 3/2 and T = 2 multiplets, will allow a better understanding of the structure of these states and may provide more detailed evidence of the nature of the isospin-dependent forces in the nucleus. 65 APPENDIX A PJ\HTICT ,li!-l~'NE!HGY DJi;•1.1F.HMIN/\'rION Jt'HOM rn•1°:C1J'HO<aV\1'H I l/\'I.' /\ . A key step in the procedure outlined in Chapter II for extracting precise reaction Q-values from the Oak Ridge and Caltech spectrograph data is the calculation of the particle energy from the measured position of the group at the focal plane. Since the two instruments used in the present work have quite different design and instrumentation, as well as diverse histories, the calculations will be discussed separately for the two devices. A. The Broad-Range Spectrograph at Oak Ridge The kinematic energy of a particle with charge Ze and mass M, bent in an arc of radius R by a uniform transverse magnetic field B, is where K is a constant determined by the units used. For B in kilogauss, R in inches, and M and E in MeV, K = l . 3132417 Since the focal plane can be both tilted and translated to adjust the focusing properties for different reactions, the orbit radius R is a complicated f'unction of the observed position Xd on the focal plane. The average radius R is calculated from the effective position Xe of a group (determined by a straight-line 66 projection of the observed position onto a nominal focal plane position, as discussed below) by means of the r e lnt:Lon (Ball im;11 ) ::J R cn xe =I n=O n For X in cm and R in inches, the coefficients C a r e: n n c 0 27 . 52 1 1. 9227 x 1 0 -l 2 - 0 . 50623 x 10- 6 3 +0 .76329 x 10- 9 The geometr y of t h e project ion to det ermine Xe f r om Xd is illust rated in Fi gur e 5. The orientation of the focal pl ane is controlled by two screw sha:rts, one of which is r i gi dly mounted to a r eference sur face . mensions n a nd n • 2 1 The or ientation is measured by the di - Us ing t he not ation of Figure 7, the effect ive position Xe i s rel ated to the observed position Xd by where 6:i_ = L(l - cos Q), ( D : n ) - [Xd cos Q + I.(1 - cor; g) - L ] tan Q 2 1 0 tnn '1' 67 lllll1 tan -1 The dimensions used for the calculations were Ll = 52.058 inches L2 = 0.87 inches 86 .92 inches L DO = . 3.074 inches 'I' = 37.5° B. The Caltech Spectrograph The data for precise Q-value measurements observed in the Caltech 61-cm-radius, double focusing spectrograph were measured by the sixteen-counter array at the focal plane. The calibration of this system and the data handling have been discussed in detail by McNally (1965 ) and Moss (1967). In brief, the counter positions and the dispersion characteristics of the instrument were combined with the measured field (NMR freq_uency) to determine an effective field f for each detector. versus effective field. Then tne data were plotted as counts The kinetic energy of a particle of mass M (units of the proton mass ) and charge Ze is determined from the effective field f by the relation ~L'h c cul:rl>rr.i:L:tnn 19(;!5). factor K lu uJ.J.ghtly J.'lldd. dqil~11dt~11·L (Ml'.N1d.l;v '.fue value of' K was calculated f'rorn the relation where !:::.. was determined by a smooth inte rpolation (using Lagrangian coef'f'icients f'or f'irst and second dif'ferences) in Table l. The constant K is defined to be the value of K f'or the 8.795 MeV ThC" 0 a-particle group. Usually the correct K for a particular data set was inferred f'rom a best f'it (in the chi-square sense) to several calibration groups. The nominal value of' K ( corresponding to the 0 observation of the a-source group at f = 27702 kH z ) is 1.138348 4 x 10-S (keV/kH ) z 2 • 69 APPENDIX B POSITION-SENSITIVE DETECTORS The construction and principles of operation of a solid-state position-sensitive detector are described in several articles (for example, Bock 1966, Kalbitzer 1967, Doehring 1968, Melzer 1968, and Kalbitzer 1970) and will not be reviewed here i n detail. The de- tectors used for the experiments discussed here were 4 or 5 cm in length, 1 cm in width, and about 400 microns in (active ) thickness. The detector provides two signals of opposite polarity. The inte- grated charge of the first signal is proportional to the energy E deposited in the detector, while that of the second signal is proportional to (~)E where L is the detector length and X is the distance lengthwise along the detector at which the event was detected. To extract the position information properly, the second signal should be divided by the first. Such arrangements have been used successfully (Bock 1966) but require some care, as well as moderately sophisticated instrumentation. If the energy E is essentially constant across the detector [(~) 2 << (~) 2 where 6P is the desired position resolution], and the group of interest produces a distinct line with little or no background in the energy spectrum, then the position spectrum can be monitored directly with a simple gating arrangement, such as the one shown schematically in Figure 6 for the measurements at Caltech. This system could be modified to store spectra for several particle 70 groups, though the ope ration grows Jncreasingly complex with the addition of more channel s. It is especially awkward ii' the experj. - ments involve frequent and substantial changes in particle energy . A more versatile approach was chosen for the work at Oak Ridge (Figure 7). The events were recorded in a two-dimensional array according to the two signals from the detector. In such an array (Figure 13) the upper edge of the PSD appears as a line (X = L). These arrays [(dimensioned 50 x 200 for Ex (PxE)] were dumped to an on-line computer where the spectra were sunrrned over fixed limits in the energy spectrum, to monitor the position spectrum during the run. The data were then stored on magnetic tape for further analysis offline. To obtain final spectra, the particle group was located in the energy spectrum (Figure 12). To "divide out" the energy depen- ence, the position spectra corresponding to the several energy channels spanning the peak and adjacent background were compressed to equal lengths (100 channels) and summed. A smooth background was subtracted in each position channel, when necessary. This process (or any equivalent procedure) is sensitive to the zero levels for both energy and position. The energy zero was checked by observing two particle groups of known energy (e.g., alpha particles and tritons) in the detector at the same time. The position zero could then be obtained from the intersection of the edge line with the energy zero. These measurements were confirmed in the (p,t) experi3 ment by pulser measurements, and in the ( p, He) by observing the two ex line s from a ThC' alpha source through o. ladder mask, o.nc1 extrupolating to the common zero point . 7l In actual operation, the position signal, when corrected for energy dependence, is not precisely proportional to distance along the detector, but rather is a smooth function of position with small quadratic and cubic dependence on X. This departure from linearity affects not only the centroid of peaks observed in the detector, but also the relative yield from channel to channel, since equal increments of pulse height (i.e., channels) at different locations along the detector span slightly different increments of position . This effect does not alter the inte- grated yields, provided the background is subtracted properly, but does produce some distortion in a spectrum that should be flat . This shape can be removed by multiplying the counts in each channel by (dX )-l which can be calculated from the nosition versus d channel ' };:' pulse-height calibration. This proce dure was used to construct the composite spectra presented in the present work (Figures 10, 1 2 , 17), but the single-run spectra ( such as Figure 11) were left uncorrected . 1 72 APPENDIX C ANALYSIS OF ANGULAR DISTRIBUTIONS FOR TWO-NUCLEON-TRANSFER REACTIONS The mechanism of the two-nucleon-transfer reactions under consideration in this woTk was assumed to be the direct transfe r of an s-state di-nucleon for the cases of interest. Several aspects of this assumption have been discussed by Glendenning (1961), Newns (1960), and Adelberger (1967a). The theory of direct reactions has been discussed rather complete ly by a number of authors (e.g., Tobocman 1961, Bassel 1962, and Satchler 1964 ), and will not be reviewed here. 3 3 The (p, He) and ( He,p) reactions transfer a (n-p) di-nucleon in either a singlet or triplet spin state; the (p,t) transfers a di-neutron in a singlet state. The selection rules for these reactions are different for the two spin states. Since the target nuclei for all the reactions in the present work were Jn = o+, T =T z = 1 nuclei, these rules can be written: ~ s = 0 ~o, 1, 2 ;Jf r: where ft is the spin of the t r ansferred di-nucleon, L is the angular momentum of the di-nucleon around the core nucleus, and Tf, Jf, nf are the isospin, spin, and parity, respectively, of the residual nucleus. Obviously the (p,t) reaction is the only one of these 73 three reactions which can populate T = 0 states, since only this reaction feeds Tz = 0 nuclei, for the target nuclei employed :In the present study. The shape of the angular distribution is determined principally by the angular momentum transfer L. If L is determi ned from the measured distribution, the selection rules can be used to interpret this information in terms of the spin and parity of the residual nucleus. For the determination of the angular momentum transfer, angular distributions for several values of L were calculated for each reaction, from the (zero-range) Distorted-Wave Born-Approxi- mation (DWBA) theory of direct reactions, by the computer code JULIE (Bass el 1966 ). These calculated distributions were subse- quently compared with the experimental results. For this calcu- lation, the transition amplitude is determine d from wave functions that are eigenfunctions of the optical plus Coulomb potentials in the entrance and exit channels, i. e ., wave functions which asymptotically describe the elastic scattering process for the channel of interest. The general form of the optical potentials used f or the calculations is d h U(r) = -V f(x) - i(W-W' dx'[f(x' )] + (me-) 2 l d Vso x dx [f(x)] l·cr --7 --7 1( X= l/3 r - r~ll \.L a ' ( ) fX (l + ex)-l = In practice , since elastic scattering has not been studied f or a numbe r of the channels under consideration, the procedure has 74 been to select potentials determined for similar channclo, and sim:i.lur energies, to thooe o:f :l.nterc ot . 'ruble l?i mmrrnur:I ~uu ·LJ1e parameters used for analysis :in this work and the sources or these para.meters. In general, these potentials were chosen as the simplest and/or most readily available potentials which produced satisfactory fits to the data. The potentials for the 14 C(p, 3He) 1 ~ distribution were taken from a study by Cosper (1967 ) of the reaction at a number of energies up to 50 MeV. 12 10 c(p,t) c The potentials for the 12 14 14 c(p,t) c and c(p, 3He) 1 ~ distributions were initially taken directly from a study by Cosper (1967 ) of the reaction 12 10 c(p,t) c at a number of incident proton energies up to 50 MeV. These data were fitted with an optical potential similar in form to the potential given above with W = 0 (i.e., a surface-absorption potential only). The parameters obtained from that study were presented as smoothly varying f'unctions of incident energy and excitation energy in the residual nucleus. It was found in the present work that the fit 3 to the (p, He ) distribution was substantially improved by assuming a volume-absorbing potential (W' = o, above). The calculated (p,t) distribution was altered slightly by a similar change in radial dependence of the absorptive potential, but in either case, the strong forward-angle peaking usually obtained for L = 0 transitions was not predicted. For the 18 2 o(He,p) °F distribution, the potentials chosen were those used by Hensley to fit ( 3He,p) distributions for reactions on 15 11 19 B, N and F. satisfactory fits for the 26 Mg ( These potentials did not produce 20 3 He ,p) Al distribution, so the potentials use d by Clark (1970) in a similar study of this reaction 75 at lO MeV incident energy, were chosen instead . No attempt was made to adjust the parameters for an optimum fit to the reactions studied, beyond the selection procedure outlined above . The bound-state wave f'unction for the di-nucleon relative to the core was calculated as an eigenstate of the potential u c - v0 ( l + e x)-1 U(r) x = (r - R)/a, R = r on Ac1/3 2 f z~cc z (3 - E_) r < R c uc = R 2 c 1z z ~ c A l/3 R = r c oc c where Z x is the charge of the di-nucleon, Z c core and Ac is the mass of the core. is the charge of the The well depth v 0 was adjusted to produce the correct binding energy for the di-nucleon. The radius and diffuseness parameters used for all the calculations in this work were r r on =l.5f, oc = 1.3 f, a 0.6 f. 76 J\PPJi!NnIX D THE QUADRATIC MASS FORMULA; COULOMB ENERGIES; AND MASS PREDICTIONS J!l.necke (1969) has discussed in some detail the derivation of' the quadratic mass formula for isobaric multiplets 2 m(A,T,T) = a(A,T) + b(A,T)Tz + c(A,T)Tz By assuming the isospin-nonconserving interaction to be of the form KL Ct - tzi)(t - t/) He = i<i where the sum extends over all pairs in the nucleus. The coef'f'icients a, b, and c can be expressed in the f'orm (using J!l.necke' s notation) a= l/2(m n 2 + m.._)A + < Ho > + E(o) - T(T + l) E( ) .tl c c b c The terms E(i) are related to the reduced matrix elements (f'rom the c Wigner~Eckhart theorem) as follows: 77 ) II H(O c E(o ) = < T c -1 E(l) = c J T(T + 1) E( 2 ) = c II T > < T II r/1) c II T > 1 VT(T + 1)(2T - l)(2T + 3) < T II H(2) c II T> The addition of charge-dependent forces arising from other sources than the Coulomb energy of tensorial r ank less than 3 will not change the form of the equation outlined above . The Coulomb force is assumed to be t he major isospin-nonconserving interaction in nuclei, and the terms E ( i ) are generally referred to as Coul omb energies . c If the masses of three members of a T = 2 multiplet have been measured, the coefficients a, b , and c can be determined immediately from the relation 1 a b 3 -2 = 1 c 2 0 0 2 -2 -1 2 1 1 m(O) m(l) m(2 ) where m(T ) is the T = 2 mass for the member with projection T • z z The Coulomb energies E(i) can then be determined from the relations c given above. Also the masses of the Tz = -1, - 2 members can be calculated from the relations (obtained immediately from above) m( -1)] ( m(-2 ) - = r3 -3 6 -8 L. m(of1 m(l ) i [ m(2)j 78 REFERENCES Aclums, J.M., A. Adamu and J.M. Culbert 19fill, .J. Pltyo. A Uerh~ n ~1 , l_, . 549. Adelberger, E. G. l967a, Ph. D. Thesis, Caltech (unpublished). Adelberger, E. and A. B. McDonald l967b, Physics Lette rs 24B, 270. Adelberger, E. G., A. B. McDonald and C. A. Barnes 1969a, Nuc. Phys. Al24, 49. Adelberger, E. G., C. L. Cocke, C. N. Davids and A. B. McDonald 1969b, Phys. Rev. Letters 22, 352. Ade]~erger, E. G., A. v. Nero and A. B. McDonald 1970, Nuc. Phys. Al43, 97~ Armini, A. J., J. w. Sunier and J. R. Richardson 1968, Phys. Rev. 165, 1194. Ajzenberg-Selove, F. and T. Lauritsen 1968, Nuc. Phys. All4, l. Ajzenberg-Selove, F. 1970, Nuc. Phys., to be published. Ball, J. B. and R. L. Aub;I..e 1968, priva te communication (ORIC computer code, Magdoc IV). Barkas, w. H. and M. J. Berger 1964, in Studies in Penetration of Charged Particles in Matter, NAS-NRC Publication 1133, page 103. Bassel, R. H., R. M. Drisko and G. R. Satchler 1962, Oak Ridge National Laboratory Report ORNL-3240 (unpublished). Bassel, R. H., R. M. Drisko and G. R. Satchler 1966, Preliminary Report on the Code JULIE (Supplement to Oak Ridge National Laboratory Report ORNL-3240) (unpublished). 79 Batusov, Yu. A., s. A. Bunyatov, B. M. Sidorov and v. A. Yarba 1967, Joint Inst. Nucl. Res. (Dub.na), Report Pl-3::J72 . Black, J. L., W. J. Caelli, D. L. Livesey and R. B. Wats on 1969 , Phys. Letters 30B, 100. Black, J. L. 1970, Private Conununication. Bloch, R., R. E. Pixley and P. Trulil 1967, Phys. Letters 25B, 215 . Boch, R., H. H. Dubm, W. Melzer, F. Puklhofer and B. Stadler 1966 , Nucl. Instr. and Meth. 41, 190. Borggreen, J., B. Elbeck and L. P. Nielsori 1963, Nucl. Instr. and Meth. 24, 1. Cerny, J., and R. H. Pehl 1964, Phys. Letters ~ 234. Cerny, J. 1968 , Ann. Rev. Nucl. Sci. 18, 27. Cerny, J., .s. w. Cosper, G. w. Butler, R.H. Pehl,F. S. Goulding, D. A. Landis and C. Detraz 1966, Phys. Rev. Letters !.§_, 469. Clark, G~ J •., P. B. Treacy and S. N. Tucker 1970, Nucl. Phys. Al43, 109. Cosper, S. w., H. Brunnader, J. Cerny and L. McGrath 1967, Phys. Letters 25B, 324. Dietrich, F. S. 1965, Nuc. Phys. ~ 49. Doehring, A., S . Kalbitzer and W. Melzer 1968 , Nucl. Instr. and Meth. 59, 40. Douglas, B. A., B. R. Gasten 1097. and A. Mukerji 1956, Can. J. Phys. ~ 00 Endt, P. M. and c. van der Leun 1967, Nuc. Phys. Al05, 1. Freeman, J. M., J. G. Jenkin and G. Murray 1966, Physl Letters 22, 177. Garvey, G. T. and I. Kelson 1966, Phys. Rev. Letters 16, 197. Garvey, G. T., w. J. Gerace, R. L. Jaffe, I. Talmi and I. Kelson 1969, Rev. Mod. Phys. 41, Sl. Glendenning, N. K. 1961, Nuc. Phys. ~ 109. Goosman, D. R. and R. w. Kavanagh 1970, Phys. Rev. £!., 1939. Groce, D. E. 1963, Ph. D. Thesis, Caltech (unpublished). Hardy, J. C., H. Brunnader and J. Cerny 1969, Phys. Rev. 183, 854. Hardy, J. c., H. Brunnader and J. Cerny 1970, Phys. Rev. £!., 561. Henley, E. M. 1969, in Isospin inNuclear Physics, edited by D. H. Wilkinson, North Holland Publishing Company, Amsterdam, page 15. Hensley, D. C., P. H. Nettles and C. A. Barnes 1968, Phys. Letters 26B, 435. Hensley, D. c. 1969, Ph. D. Thesis, Caltech (unpublished). Hensley, D. C. and P. H. Nettles 1970, Private Communication. JMnecke, J. 1969, in Isospin in Nuclear Physics, edited by D. H. Wilkinson, North Holland Publishing Company, Amsterdam, page 297. Kalbitzer, S. and W. Melzer 1967, Nucl. Instr. and Meth. 56, 301. Kalbitzer, s. and W. Stumpfi 1970, Nucl. Inst. and Meth. 7.J..., 300. Kelson, I. and G. T. Barvey 1966, Phys. Letters 23, 689. Kingston, F. G., R. J. Griffiths, A. R. Johnston, W. R. Gibson and E. A. McClatchie 1966, Phys. Letters 22, 458. 81 Kuan, H. M. D. W. Heikkinen, K. A. Snover, F . Riess and S . S . Hanna 1967, Phys. Letters 25B, 217. Lee , L. L., J. P. Schiffer, B. Zeidman, G. R. Satchler, R. M. Drisko and R. H. Bassel 1964, Phys. Rev. 1 36, B971 . Lynch. B., G. M. Griffiths and T. Laur itsen 1965, Nuc. Phys . 65 , 641 . Matous, G. M., G. H. Herling and E. A. Walicki 1966, Phys, Rev. 152, 908. Mattauch, J. H. E., w. Thiels and A. P. Wapstra 1965, Nuc . Phys. 67, 1. R. L., J. C. Hardy and J. Cerny 1 968, Phys. Letters 27B, 443 . McGrath, R. L. 1 969, in Nuclear Isospin, edited by J. D. Ande rson, s. D. Boom, J. Cerny and W. W. True, Academic Pr ess (New York and London), page 29 . 1968 , Ph. D. Thesis, Caltech (unpublished). cDonald, A. B. 1969 , Ph . D. Thesis, Calte ch (unpublished). cNally, J. H. 1966, Ph. D. Thesis, Caltech (unpublished). Nettles, P. H., D. C. Hensley and T. Tombrello 1969, in Nuclear Isospin, edited by J. D. Anderson, S . D. Bloom, J. Cerny a nd W. W. True, Academic Press (New York and London ), page 819 . Newns , H. C. 1960, Proc. Phys. Soc. 76, 489 . Olsen, D. K., w. D. Harrison, R. E. Brown and C. A. Barnes 1968, Annual Report C00-1265-67, John H. Williams Laboratory of Nuclear Physics, University of Minnesota, Minneapolis (unpublished) , page 83. Poskanzer, A. M. , P . L. Reeder and I Dostrovsky 1965, Phys . Rev . 138, Bl8 . 82 Richards, H. T. 1960, in Nuclear Spectroscopy (Part A), edited 1Jy F. Ajzenberg-Selove, Academic Press (Ne w Yorl<), page 9DF . Riess, F ., W. ,J . O'Connell, D. W. Hej.kkincn, U. M . Ku.an 1md ~; . fi. Hnrn111 19,6 7, Phys . Rev . Letters 19, 367 . Satchler, G. R. 1964, Nuc . Phys . ~ 1. Snover, K. A. , D. W. Heikkinen, F. Riess, H. M. Kuan and S . S. Hanna 1969, Phys. Rev . Letters 22, 239 . Tobocman, w. 1961, Theory of Direct Nuclear Reactions, Oxford University Press (London). Weinberg, s. and s. B. Treiman 1 958, Phys. Rev. 116, 465 . Wilkinson, D. H. 1964a, Phys . Letters 11, 243. . Wilkinson, D• H. 1964b, Phys . Letters g 348 . Wilkinson, D. H. 1964c, Phys. Rev. Letters 5 571. Wigner, E. P . . 1957, in Proc. Robert A. Wel ch Found • Conf. on Chemical Research, Houston, Texas, edited by W. O. Milligan (the Robert A. Welch Foundation, Houston, Texas) , Vol. I, page 88. 83 F'ractional Changes in S pectrometer Culil>ration Factor with Magnetic Field f'(MHz) t:.(x l0- 3 ) l4 . 24 l6 .05 l8 -.30 20 -.52 22 -. 60 24 -. 52 26 -. 37 28 o.o 30 .57 32 l.22 34 l.96 36 2.80 38 3 . 73 40 4.77 42 5.80 44 6.90 84 Excitation Energies Predicted for the Lowest T = 2 States in l 2c and l~ (entries in MeV) l2c l~ Based On Reference 27 .58 l2.47 T = 3/2 levels in JH.necke l969, A = ll, l3 < 31.3 < l6.2 Upper limit for Adelberger l970 Cerny l968 m(l2:se) 27 .98 l2.87 G.K. prediction for Garvey l966 m(l~e) 27.5 ± .l l2.67 ± .07 Experiment Cerny l968 U5 TABLE 3 14c Target Thickness/Composition Measurements from 12-MeV :Material Method(a) Au( thick) Width · Au(thin) Yield (12C+l4C) Shift 12c Yield(e) y 3 He Scattering at 90° /y (b) -1.L..:!:.Q (c) 6.E (keV) ~ff 99.2 10.45 0 . 204 275 56 389 1. 73 1.41 375 (d) 62 14 (f) c 284 a. See Section II-C. b. Yield Ratio, for yield method only. c. Average stopping power (units l0- 15 ev-cm 2 ) for incident and exiting energy. d. Equivalent 12 c thickness, A= Ci~) (1' 14 use in energy loss calculation. The of the form (CH) , from which the factor n e. f. assumed to be Ee + EH EC was obtained . Thickness compared to previously measured l2c target, A = 44 µg/cm 2 14 12 Final c thickness, corr ected for c and H content . 4. 4377 15 . 226 27 . 595 56. 9 44 . 0 Average 8 . 4512 7. 7500 7. 7821 7. 7251 7.4201 8 . 6723 lo lo lo lo lo 40 20°/20° 20/20° 20/20 20/20 20/20 30°/20° (continued) Focal-plane orientation (D and D } defined in Appendix A. 1 2 12C(T = 2) 13c(l5 . 112) (e) 1 ~(4. 444)(d) 11 B(O . O) b. +.0007 +. 0011 - . 0014 27.5911 27.5911 Target angle (Qt) is the angle between the target normal and the incident -beam momentur::. 81. 7 36 . 0 59 . 1 45 . 5 52. 0 12 c(T = 2) Identification a. 4. 951 5 . 357 4. 529 4. 802 8. 3902 40 27 . 6019 20°/20° 58.7 B. 3101 27 . 5973 40 57 . 7 (MeV) 25°/20° 4.951 5 . 351 (f . s . = 100) (inches) (kG) Excitation 8 . 2307 Channel(c) ~1/£2 (b) Field 40 , Aperture 30° /20° (Qm/Qt ) Angle(a) Measurement of the Exci tation Energy of the Lowest 12 T = 2 State in c TABLE 4 (J) (J) See discussion on page 26 of the use of this group for calibration. Hensley 1969. Ajzenberg-Selove 1968. d. e. See Figure 18 for channel-to-position conversion. c. TABLE 4 (continued) CD --l See page 30 for a discussion of the entries in this column. The measurements at 20° with the 4° aperture were made with a different focal plane orientatic~ from that of the measurements made with the 1° aperture. f. 78 e. 82.4±8.0 29 The total resolution is calculated as the quadratic sum of the contributions listed. 88 32.4±5.0 76.5 d. 47 50 76.5±5.0 (keV) The focusing of the beam by the last quadrupole before the target chamber gives rise to a spreac ~n the incident angles of bombardment and a resulting kinematic shift. 72(f) 47 104 (keV) c. 18 .5 81 (keV) Observed Revised Tctal(e) This contribution to the resolution arises from the improper orientation of the focal plane ::Or the particular group being observed, resulting in a residual kinematic broadening. 40 18 42 (keV) Total(d) b. 20° 12 c(27.59) lo 51 (keV) Quadrupole(c) 12 14 c and c(p,a)~ Measurements(a) Focal Plane(b) c(p,t) The entries in this table are expressed in laboratory particle energy. After the observed ~"idth was corrected for the experimental resolution, the result was re-expressed in 12c excitatio~ energy. 20° 12 c(27 .59) lo (keV) Target 14 a. 20 0 Angle Aperture 11B( g.s. ) State Width Analysis for TABLE 5 0 Cl 6.7671 5 . 7689 20 80/80 80/80 5 . 5331 40 30°/15° 6. 7685 5 . 6000 40 25° /15° 80/80 5 . 7817 20 60/60 5 . 7689 5 . 7519 40 10°/ 10° 80/80 5 . 6601 40 20/0° 3. 653 3. 748 4.726 5 . 096 · 4. 498 4. 799 52. 5 51. 6 56 . 6 56 . 6 60.3 59 . 5 57. 4 58. 8 61. 2 61.5 (f . s .= 100). Channel( c) (continued) 3. 551 3. 623 . 3. 757 3. 877 4.171 4. 412 (inches) (kG) 5. 6600 D /D (b) - 1- 2 Fiel d 40 Aperture 20°/0° (Qm/Qt) Angle(a) Measurements of Excitation Energies of -- TABLE 6 1 - 0. 0008 -0 . 0001 12. 7160 12. 7160 12. 7133 12. 6958 12. 7134 12. 6925 12. 7107 12. 7080 (MeV) Excitation 2s States 2s(T = 2) 12s(g.s . ) 1 I dentification en r.o 6. 7671 80/80 See Figure 24 for channel-to -position conversion. Ajzenberg-Selove 1968 . Ajzenberg-Selove 1970. d. e. ~(T = 2) 1 c. ~(14. 86 ) 1 Focal plane orientation (D and D ) defined in Appendix A. 2 1 12. 710 14.860 1 ~(2.313) (e) b. Average of the 12.7-MeV-State Measurements 60 . 0 2.3056 1 2:s(o. 950)(d) I dentification Target angle (Qt) is the angle between target normal and inci dent-beam momentum . 40 85 . 0 . 948 (MeV) ( f . s . = 100) 43. 8 Excitation Channel (c) a. 10 °/10° 3. 757 3. 877 6.7126 80/80 5.5619 (inches) (kG) (l:,im/'\) Aperture D /.:2 (b) -1 2 Angle(a) Field Measurements of Excitation Energies of 1 2:s States TABLE 6 (continued) 0 co 57 4 10 14. 86 The focusing of the beam by the last quadrupole before the target· chamber gives rise to a spread in the incident angle of bombardment and a resulti ng kinematic shi~. The total resolution is calculated as the quadratic sum of the contributions listed. See page 36 for a discussion of this term . c. d. e. 68 This contribution to the resolution arises from improper orientation of the focal plane for the particular group being observed, resulting in a residual kinematic broadening. 228±30 76 74.5 81 (keV) f-' rj) Revised Total(e) b. 60 81±10 79±10 81±5 (keV) Observed All entries in this table for widths are expressed in particle energy in the laboratory. 20 62 60 28 30 53 (keV) Total(d) 32 (keV) Quadru~le{c) a. 0 0 54 4 10 12. 71 0 54 2 8 12. 71 (keV) Focal Plane(a) 14 (keV) Target 40 20 AJ2erture 80 Angle o.o (MeV) State Width Analysis for the 14c(p, 3He) 1 2.s Measurements(a) TABLE 7 92 TABLE 8 Predictions of the Mass of' 1 2ne -~(a ) T 1 m( 2ne) Source (b ) (MeV) (MeV ) 1 3 . 968 25.503 T=l, A=l 2 mass difference(c) 2 4 . 042 25.577 T=l , A=lO, 14 mass differences(d) 2 4 . 014 25.549 ~' A=9, 11, 13, 15 mass differences 25 . 0 Garvey-Kelson prediction(e) 25 . l Estimat ed beta- decay endpoint energy, from the measured(lifetime and calcul ated ~ value fJ <28 . 3 Upper l imit from beta- decay endpoint(g) a. The quant i ties ~(T) and T were defined on page 39 of the text . b. All masses not otherwise referenced were taken from Matt.auch (1 965) . c. d. The 1 ~ mass was obtained from Adams (1968) . The 1 0 c mass was obtained f r om Freeman (1966) . e. Garvey 1966 . f. Poskanzer 1965 . g. Cerny 1 968 . 93 TABLE 9 Pred:Lctions and Previous Meo.suremento or the Exc:l.tat:i rn1 2 Energy of' the Lowest •r = 2 State o1' °F Excitation Source or InEut References (MeV) 6.59 200 6.48 20 Ne(T=2); ' 20Na-20F a,b a,b,c (20Na _ 20F) 6.43 ± 0.1 (Experiment) d 6.523 ± 0.0:55 Experiment e a. Mattauch 1965. b. The ~ONa mass was taken from Endt 1967. c. Adelberger 1967a, b; Bloch 1967, Kuan 1967. d. Cerny 1964. e. Hardy 1969. 94 TABLE 10 Measurements of the Excitation Energy 2 States in 20F(a) of T == Angle Frequency (b) Uncertaint~(d) Excitation(c) Nucleus Magnet/target (kHz) (keV) (keV) 10°/10° 34775 -0.163 2. 4 18F 20°/20° 34626 +0 . 267 3.2 1 8F 10°/10° 32547 6514.0 2. 2 20F 20°/20° 32415 6512 . 5 2.2 20F 10°/10° 30053 8211.9 3. 4 20F 20°/20° 29923 8208.7 3. 4 20F a. The results of these measurements are summarized on page 46 . b. The tabluated frequency is the effective NMR frequency, defined in Appendix A, at the centroid of the peak . c. All nuclear masses used in this analysis were taken from the current tabulations (Mattauch 1965). d. This column gives the contribution to the probable error from the estimated uncertainty in the frequency . analysis is discussed on page 45 . The complete error TABLE: l l Predictions and Previous Measurements of the Excitation 28 Energy of the Lowest T = 2 State in Al Source Excitation Reference (MeV) 6 .06 28Mg, 2\ _28Al a,b 5.983 ± 0.025 . Experiment c [ 5.989 ± ?] Experiment d a. Mattauch 1965. b. The c. Hardy 1970. d. Clark 1970. 28 P mass taken from Endt 1967. 9G TABLE 12 Measurem~nts of the Excitation Energies of States in 28Al(a) Uncertaint/b) Identification Angle Frequency Excitation (magnet/target) (kHz) (keV) (keV) 50/50 35310 5995.4 3.0 10°/10° 35283 5997.9 3.0 Lowest T=2 28 State of Al 15°/15° 35245 5996.6 3.0 (protons) l5°/l5° 33279 7490.7 lO 20°/10° 33223 7491.8 lO 1st-Excited T=2 28 State of Al 30°/15 33071 7490.6 lO (protons) 15°/15° 33335 7449.5 lO 20°/10° 33284 7446.5 lO 50 /50 37030 6813 l.5 10°/10° 36979 6813 l.5 Deuterons 27 Al(T=3/2) 15°/15° 36899 6812 l.5 E =6815±2 keV(c) 20°/20° 36780 6814 l.5 l0°/l0° 42739 l0°/l0° 42215 2.0 1065 Second member of doublet near 7.5 MeV excitation x 3 25 (Alphas) Mg ground-state 3 26 (tritons). Al:, 1059 kev{c) a. The results of these measurements are surmnarized on page 53. b. The tabulated uncertainty is the contribution to the total uncertainty resulting from the determination of the centr8.l frequency only. The complete analysis of probable errors for the 28Al states is discussed on page 53. c. Endt 1967. C + p, 50.5 MeV 1.25 1. 77 1.25 1.81 1.25 1.25 Cosper 1967. Hensley 1969. Matous 1966. Lee 1964. Clark 1970. c. d. e. f. g. (See also the discussion in Appendix c.) The Coulomb radius para.meters r C = 1. 25 f f or all channels. 0.45 0.30 0 . 65 0.829 0.60 0. 40 1.25 b. 1.25 1.08 1.2 1.05 1.25 1.25 0.60 1.25 r'0 (f) These parameters are defined in Appendix C. 12. 0 12.0 0 23.2 35.9 21. 7 1.25 - 0.40 a (f) a. 63.7 28Al + p 64.3 20F + p 153.0 183.3 180 + 3He 26Mg + 3He 155.0 53.3 171.29 1.25 ro (f) - (MeV) (MeV) 59. 74 w v 12:8 + 3He 14 C + p, 63. 4 MeV 12c + t 14 Channel - 8 - - - - - 44 28.83 29.11 (MeV) W' - - - vso (MeV) In all cases W = O. so 0.47 0.60 0.45 0.592 0.60 0.40 0.60 0.40 (f) a' Optical Potentials(a),(b) Used for Analys~s of Angular Distributions TABLE 13 ! ./ g g d,f d,e c c c c Reference (.!) --..l 98 TABLE 14 Summary of Masses for the Lowest •r = 2 Mult:Lplets in fl. = 4n Nuclej_ (u) A T z= 0 -- T l - z=- - 2 - z=- - 8 32427 ± lO(b) 27483 ± lO ? 31600 ± ll5(c) 12 27595 ± 20(d ) 27595 ± 20 26080 ± 20(d) 127l0 (< 28 300 ( e ) ) 16 l798l ± 8(f) 227l7 ± 8 15613 ± 8(g ) 9928 ± 7 13693 ± l 6(h) 20 9690.5 ± 2.8(i) l6732 ± 2.8 6501 ± 3(j ) 6513 ± 5 3799 ± 8 (h) 24 1503 ± 5(k) 15436 ± 5 -2450 ± lO(l) 5968 ± lO -5949 ± lO(h) 28 -6269 ± 5 l522l ± 5 (m) -10859 ± 4.4(n) 5997 ± 5.5 -15020 ± 6 (h) a. b. c. d. e. f. g. h. i. j. k. l. m. n. T Each entry gives the (T=2) atomic mass excess/excitation energy, in keV. Black 1969. Cerny 1966, Batusov 1967. Present work; Cerny 1968. Poskanzer 1965 . Adelberger 1970. Hensley 1968. Mattauch 1965. Adelberger l967a,b; Bloch 1967; Kuan 1967; Hardy 1969. Present work; Hardy 1969. Adelberger l967b, Riess 1967, Hardy 1969. Hardy 1969, Hensley 1970. Hardy l970, Snover l969. Present work; Hardy 1970. - 3433 ± 8 -4180 ± 22 -4805 ± 12 9690 ± 3 1503 ± 5 - 6269 ± 5 20 24 28 81 ± 2 76 ± 4 4216 ± 8 4962 ± 22 5587 ± 12 244 ± 5 227 ± ll 215 ± 6 72 ± 2 75 ± 4 3375 ± 20 224 ± 11 97 ± 32 E (2) c (keV) 163 ± 11 [1771 ± 50] E (1) c (keV) [2787 ± 60] 489 ± 34 288 ± 96 (keV) c 1 For A = 12, b was calculated from the average predicted mass of ~e from Table 8 . The predicti on s chemes are discussed on pages 38-39. the c for T = 2. The b coefficient for A = 8 was taken from t he T = 1 mul t iplet, and used to extract -2592 ± 20 17981 ± 8 16 a. [-2004 ± 60] 27595 ± 20 12(a) [ -988 .5 ± 50] (keV) (keV) 32427 ± 10 b a 8 (a) A Mass E~uation Coefficients and Coulomb Energies for A = 4n, T = 2 Multiplets TABLE 15 (.!) (.!) 100 TABLE 1 6 Mass and Stability Predictions for Tz Nuclide Mass Excess(a) (keV) -2 Nuclei Proton Threshold (2p~ Threshold (keV) (keV) Stab il.i t;y: ( c) 120 33559 ± 250 30280 ± 2 unbound l6Ne 24061 ± 88 22586 ± .5 unbound 20Mg l7532 ± 38 20263 ± 70 19897 ± 5 bound 24Si l0771 ± 90 14055 ± 80 l4l99 ± 25 bound 28s 4198 ± 50 (7136)(b) 74l0 ± 15 bound a. The quoted uncertainty is from experimental uncertainties in masses for T b. z O, 1, and 2 members of the multiplet, only . 27 27 The mass of p was predicted from Mg and the lowest T = 3/2 27 27 states in Al and the ground-state mass difference for s i and 27 Al, according to a procedure analogous to that outlined j_n Cha pter III for T = 2 states. c. 1"1he stability indi cated refers to decay by prompt heavy-particle emission. The bound nuclides indicate d are expecte d to be ~+-delayed proton emitters. -1250 ± 21 28p 5902 ± 23 5810 ± 97 [1718] 9605 ± 25 16567 ± 25 2405 10162 1 6492 19755 bound bound bound unbound unbound Stability The stability indicated is thus that relative to isospin-allowed Ground-state masses from Adams 1968 . Ground-state mass from Armini 1968. e, heavy-particle decay channels . and 2p + (A-2, T = 1) . The thresholds indicated are the l owest isospin allowed channels, i . e . , p + (A-1, T = 3/2) d. c. tabulations (Mattauch 1965) . b. 5910 ± 35 24Al 6505 ± 42(e) - 28370 (keV) 2p Threshold(c) masses of T = O, 1, and 2 members of the multi plet, only. z Unless otherwise indicated, t hese are based on the ground- state masses given in r ecent 13368 ± 15 20Na 10104 ± 39 (d) - (keV) 12 Threshold(c) The uncertainties indicated are contri butions from experimental uncertainties for the 20797 ± 36 16F 12750 ± lOO(d) (keV) Excitation (b) = -1 Nuclei a. 30088 ± 100 (keV) T = 2 Mass Excess(a) l~ Nuclide z in A = 4n, T Mass , Excitation and Stability Predictions for the Lowest T = 2 States TABLE 17 0 I-' I-' For In the Caltech spectrograph, the trajectory is perpendicular to the reaction plane. The position-sensitive detector. See page 3 for further discussion. detector was either one of an array of 16 detectors (Caltech measurements only) or a 5-cm-long, shown. the Oak Ridge spectrograph, the particle trajectory in the spectrograph is in the reaction plane, as The target orientation was specified by the angle Qt• Reaction products were analyzed in a magnetic spectrograph positioned at a known angle Q m relative to the incident-beam momentum. detector. A schematic diagram indicating the configuration of the target, spectrograph and particle FIGURE l I-' 0 I\) 103 Figure 1 104 FIGURE 2 Spectra of 12-MeV the gold backing of a 3 He elastically scattered at 90° from 26 Mg target for two target orientations. The shi~ in the high-energy edge of the group provides a measure of the Mg-layer thickness. The "Width of the group in the upper orientation measures the gold-layer thickness. further discussion. See page 8 for 105 Figur e 2 eoooo.------..-----..-----------.....----......----...,.------. ELASTIC 3He FROM Au 80000 20000 60000 + 20000 .9 NMR FREQUENCY <MHZ) 106 FIGURE 3 3 Spectra of 12-MeV He elastically scattered at 90° from the 26 26 Mg layer of gold-backed Mg target for two target orien- tations. The shi~ in the high-energy edge of the group measures the thickness of the gold layer . upper orientation measures the for further discussion. The width of the group in the 26 Mg-layer thickness. See page 8 1 07 500 F igure 3 .------P------.-------.------..-----..,..-----T------. + ELA.5TIC 3He FROM 26Mg en ....... + + + + •• 300 z ::::> 0 u 200 100 + ++ en ....... 300 z ::::> 0 u 200 100 NMR FREQUENCY (MHZ) 108 FIGURE 4 Spectra of 12-MeV · a gold 1 ayers in 14c t 3 He elastically scattered at 90° from · t a t _ions. · arget f or t 'WO t arge t orien The w.idth of th~ higher-energy group in the upper orientation measures the thickness of the thicker gold layer. A comparison of the yields of the higher-energy groups for the two configurations, gives the relative thicknesses of the two gold layers. The separation in energy between the two groups is a measure of the 14 . C-enriched-polymer thickness. See page 8 for further discuss ion. 1 09 Figure 4 12000 10000 ELASTIC 3tie FROM Au 8000 14c ~ Au c::J U') ~ z ::> 8000 c u llOOO + + 3000 UIOO .. .... + ........ + ........... ~ NMR FREQUENCY (MHZ) .. .; particle kinetic energy are calculated. See pages 65 - 66 for further discussion . t he particle trajectory to nominal focal-plane position X , from which the orbit radius e and The position Xd at which a particle is detected is projected along a straight-line extension of Geometric details at the focal plane of the Oak Ridge spectrograph (not drawn t o scale ) . FIGURE 5 I-' I-' 0 PIVOT f. 1. FIXED SCREW SHAFT Do 1 I- 2 . I . 0 I- L L1 A2 ;>L. \ POSITION . 0 Xe= Xct cose + A 1 + A 2 Xe I + f L2~ •1 --l 1-- xd ______--\ REFERENCE SURFACE POSmON FOCAL- PLANE NOMINAL FOCAL PLANE TRAJECTORY PARTICLE (..- : r. ~ ~ ....... ~ I-' I-' I-' detector at Caltech. See page 69 for a discussion of this data- acquisition method. Schematic diagram showing the simple gating network used with the position-sensitive FIGURE 6 I\) I-' I-' 113 Figure G "- ~ ~ "- ~s c co :..:J <!) () Cf) c. E > co Q) Cl c. E <( <( c. Q. E co e . a.. > f2> Q) c w E ~ a.. c .Q ~ The coincident position signal selects the appropriate half of the memory In this arrangement, the two energy signals are mixed and fed to fer each detector . See page 70 for further discussion. fer storage and the two pulse heights are recorded in a 50 x 200 (energy x position) array, a co!IT!on ADC. detectors at Oak Ridge . Schematic diagram of the electronic network used with the position-sensitive FIGURE 7 ~ I-' I-' - - ... .... IP. El2 E2 E1 IP. El1 --- PREAMP AMP AMP AMP - AMP PREAMP ...._. PREAMP PREAMP i--- - - SCA SCA - --OR STROBED AMP [_ STROBED AMP AMP STllOIED y X2 2 x 50 x 200 CHANNEL ANALYZER x, . LIHK COMPUTER -..J 'i CD ~ '=.! I-'· CJl I-' I-' The groups identified as tritons and deuterons are indicated. page 23 for further discussion. experi- See The smoothly varying 14 12 c(p,t) c background is associated with the elastic proton group, which was on the detector. ment. Energy spectrum from the position-sensitive detector, for the FIGURE 8 i-' ,- ll7 Figure 8 • • • • •• •• + • • + • • • •+ • ~ ~l ~t ~w ~ c: . .9 ·c: I- "' (/') ....J UJ z z + a: + ~ u + + + '\J >- + + <.!> oc UJ z + + · ~ s ~ 1.1...1 *™+ + + + • ......~~~.a...~~~.a...~~~.a...~~~.....~~~.....~~~~o 0 ~·· --~~~ ~ - - ~ SlNnClJ experiment. For compactness, See page 23 for f\Uth:!r discussion. The deuteron line is suppressec The broad feature on the le~ of the array the high-energy tail associated with the elastic protons. as a horizontal line at about channel 29. cy the logarithmic scale. ~s see~ The tritons are diagonal cutoff on the right is produced by the edge· of the detector. 12 7~e c(p,t) data are displayed logarithmically (o -?blank, 1 -71, 2-3 -7 2, 4-7 -7 3, 8-15 -7 4, etc. ) . 14 tt~ A typical array of raw data from the FIGURE 9 CD I-' I-' z w w >D a: Or 10 20 30 YO so I ~55566S555551llllllll423ll333333333333333223222223223332 ijQ sn77666655ij1 "66655Wl&3 ~nssG66666551il2 356666666G6Ei553 3SS6566666S6SS5 1 P~SITI~N --80 3566665656655555555555SSIU ~5566653 120 X ENERGY 21l5666666551lll31.1333331.1"11333333333321l23332333333 3S6666665514113531.131.131.1331.11lll331.13331.133333ij3331&32 ij5666666555ijllllt!llijllllll5145141l331WllllllllllWWllll 1 36676S66665555556565555$5555141l55665113 ij667766666SS66666666666666555666653 ssn6666S666S6666666676666666662 160 14c(p,t)12c I 21 12 11 1 22 ~ 1 1 121111 1 1 I I 2 111 1111131 112 1111111 1 12 1 1 1 111 2212 12 11 2122 11212 1 l 1 1 1 21121212 222 2112122 12211 2 1 212222121221 11221 2122 1 1 1 1 1 l 1 132 22 122122 2112112122121 11 1 1 . . 11 11 122322211222 212122 212222 1 1 1 1 . 1122223212121122121112222111 1 .2 1 1 1 2223322233232232222311112222 111 11 · 1 1 11323223231232333 12313222121 21 1 1 1 1 1 232333323mn2721232222 11 11 1 1 111123323323332231321233331121121111 1 1 1 1 21233333333333332222232211 122 l 1 1 1 1 1 1 1 11111~3333133233322332332222211 1 1 1 2 11 23333311331.13333323333222221211 2 l l l 112231l3231lllllll332323333333222222 l 11 l 1. 1 1 1113333ij3331&33333331l333332121 22 11111 11 1 1 1 1 11 1 1 4!213333ij311!Wll31.13331l3333222221111 2 11111 1 1 11 1 2 2 1 1 l21111&331llll&l!IWlllWl433331t332113 2 2 211 1 11 1 1 2 1 12222133333331.11t!Wll52 l21lll5115141l5145145141llt!Wllllll331.13331.11&31.133331lijij5555555111&1131.11Wll331.11W31.11Wllll331&33322122213121 231l$55555555555~1&1111511551ll&SSWIWl&l&lll&llll31.1312321 2 1 1 .1 l 1121 223311111l5145141lllllllll31.1111132212221 2 11 11 121 1 1112 12 1 1 1 1 1 11 3'&1W$55551l51411111411'13332212221 1 12221 2 11 111 1 1 22 12 11 1 121121 331.155551J514ijllllllllll33323 11 11 222 21121 111 2111 2 12 2 1 2111 131.lll51455551411111lllllll33332321111212111212 222 221221111 12 1 21 12111 31WSSSSSSSSSWSl41l323222122231212232 11 12211112 21 3112 1 12 23Ll~5""1&3333122322331223133322Z322 2221222221 1122 ;iijS555555SS55ijll3333123232'3222323233332322 l 322Z21 l 211 21 31.15556S55555141W333232223'23223222322223222322211222122 1 200 j ~ I JI I I (.() ro '1 s:: I-'• ~ lzj ~ (.() 12 14 c(p, t ) c reaction . The two c is · in · d'icat e d • 12 constructi on. See page 23 for further discussion . 12 c target of similar The insert at the top shows a triton spectrum, also uncorrected The triton group identified as the lowest T = 2 state for background contribution under the triton line, from a · in subtracted out with some care . trapezoidal peak in each spectrum is the result of the elastic protons, and can be The No attempt has been made to subtract the background contributi on from under the triton line . tvro position- sensitive detectors at the focal pl ane of the spectrograph. spectra shown correspond to the two sets of data collected simultaneously from the A composite triton spectrum at 30° from the FIGURE 10 f\) f--' 0 100 :::::> z ([) 75 .._ 01 25 u so EJ -,,... * + + : ~· ,,,.•Z\t:;:.¥1\~;+: \~.· ..,...., •i. * .. + + + + + ~ 12c ~ t •fj t-vlt "+* ~#.-..- +:\.1'+··" + +++ * .......... tt( ... ~~..•• I rt... + + + + + ••• + 'ltiif ++ j 1 )++ ,,_j+I(+ ++\+...-+ .flt +I+++*+++++ + +• ++ + ++ ++ ++ + + + + + ++ • + + + + 4++ + + 0 100 I- I f t + "' + + + + +•+ I + ++: +.:+ 30• ••·.,,.:+:•• 14c1p~t1 2c 1 - ... .. 20 + + + + + ~,. ... + + + ++ + •+ ++ \ + +t + + + + +. + + : : , · ,..~·tt ++ ~..... + ~ ++ + ... + + + ~ .... + + ++•.: ~ + + + + ++ +4 ++'\ "' + + ++ + + PARTICLE ENERGY CMEV) + + \ *~t • l.V++ ++ + ~ • ~. + ... + + ++W+ +. + I ++ +++++ ++ ++ +, + + +.; * .+ + 12<:(27.59) elastic P + + 01--~~~~~~-t-~~~~~~-t~~~~~~~t--~~~~~~-t-~~~~~~---l1--~~~~~----i 25 EJ U SC :::::> z ([) .._ 75 ~ 100 I- 125 I I-' 0 CD Ii ~ l:j I-'· {JQ ~ I-' The elastic proton group previously noted adjacent 14 12 c(p,t) c extracted triton spectrum (see Figur es 8 and 9) . See page 24 for additional discussion. of the smoothly varying background (in particle energy) under the triton line .from the final to the triton group of interest was suppressed in the spectrum by removing the contribution measurements, at 25° in the laboratory. A typical triton spectrum accumulated in a single run for the FIGURE ll :.; ,__, 123 Figure 11 .... +• •+ + "' + *· + + +• + +. + + •+ + + + ..,. + ,...... + + ·"' en _J w z z + a: I: LI "'-/ + +• + + + + + + + + + +• + + + + + + + *+ + +• + •• - tCJ - 8 + •• + z E) ....... ~ ....... en E) Cl... + + + 0 14 12 c(p,t) c experiment. The two spectra shown correspond to the T = 2 state. The bracket indicates the expected location of See page 24 for further discussion. at the focal plane of the spectrograph. the two sets of data collected simultaneously from the twc position-sensitive detectors 10° in the lab, for the A composite triton spectrum showing the scan over triton energy measured at FIGURE 12 .p.. f\) f-' 1 25 I I I I l~:lgure I I -i 12 - -i N + + ·"' ++ *+ ++ ++ +}. + •.. + + t+.i s. + +-P" + ~ ++ + + -- -~ ·~ •++ ++ + + ++ ++ ++ + + "'-+ ++;i +* \ *.t+ + ._ \i: ci ~ ~ +~·+ •++ ~. j 0 * -- ·~ *· ~ ... 5+ +-P" •-ti. + .,. + -~ LO g,, w '\+• + 't + + ++ ++ >- + .....++ ~ u ,___, + + ++ ~ -"-- -~ I - ~..... +.+I_. a: a: ·~ .~ CL + + ++ ++ .,. ;!-+ + a: w z w w _J l. + .,. ++ + + + + ++ + + ++ + + + + _._ \+ _ ir- ++ + + +t...~ .t+ ++ "'"'° •• +1;...._ +!l .~ + +*~+ +.; ~ N I I 8N -SlNnrJJ~ I 8 + I ~ :L ~ ~t+ ~[ II w '-' ++ !.-++ ++ ++ I- + " > 0 I I ~ -SlNnrJJ~ I 8 • ~ 0 the detector. 27 c(p,a)~ are shown, Al(p,a) 2~ which was used for determini ng the endpoint of See pages 24-25 for additional discussion. along with a spectrum from 14 14 12 c(p,t) c measurements . Three observations of the ground-state alpha-particle group from Typical alpha-particle spectra for calibration of the FIGURE 13 I\) CJ) !-' au • 4 f 1 1M111111111 111iff1111111111111111f&, POSITION <CHANNELS) 0,11111111 I ltt '*11'*81111 . z 0 s I- 8,J z ::> c u 50 I I l l * 11"1a ,,., . zl 25 IS -I z ::> ID u 50 15 I I- 11a,, ""' (f) I- 100 I ·(f) au 1 125 (/) 15 ALPHAS 20 DEGREES POSITION <CHANNELS) ... A 11a,, I I- "" I 20 ... " I ALPHAS 20 DEGREE: 100 125 0 25 ::> c u 50 z I- (/') 15 ::r I •• •• • • + • I - ; + • + POSITION <CHANNELS) • • • • .... + ....... .; .... • •• •• + + + + -- ne. . ··" .. ...,, . • + ~ ALPHAS FROM ALUMINUM I - POSITION <CHANNELS) • . ••• ....••..•• • • ••• · · ~ • • I -· ALPHAS 20 DEGREES --- i-' I\) (.,,' !-' ro ~ ;:: ~ '=:l ,...,. --.J The smooth curve was constructed by The calibration data See page 25 for additional discussion. 14 11 c(p, a ) B measurements are shown. least-squares analysis of these data. obtained from 14 12 (100 channels = detector length) for the c(p,t) c experiment. Calibration curve to determine focal-plane position from measured pulse height FIGURE 14 OJ I\) !-' n.. C> en lCliJr 107 ~ 109 ~ ~ C> z "" 2:: u,u .._, 113 • 11k ilk PULSE HEIGHT <CHANNELS) 119 · 119 119 14c(p,t)12c POSITION CALIBRATION a 119 aL us.---------.....--------.---------..---------..-.--------. f\) '-rJ II>- I-' (!) 1-j c CJtl I-'• (.!) I-' 1 30 FIGURE 15 Angular distribution measurements for the reaction. 14 c(p,t) 12c The vertical bars indicate estimated uncertainties for the data points. The horizontal bars indicated the spectro- graph aperture in the center-of-mass. The data points at 12° and 17° are estimated upper limits; the group of interest was not distinguished above background and continuum yield at these angles. See page 28 for a dditional discussion. F:l.e;11 re l!j 14c(p,t)12c E,=50.5 r.AeV z ~20 tu LLJ en en en L=O 0 a: U10 C.M. ANGLE <DEGREES> Deuterons at this field 14 3 1 c(p, He) 2.B lost among the other low-energy events. See page 32 for further discussion. setting were penetrating through the active region of the detector, and these events are experiment. 3 The He++ and alpha-particle groups are indicated. Energy spectrum from the position-sensitive detector, for the FIGURE 16 Vl I\) I-' 1 33 "' en + u + + + + + '-J ><.!> a: + LaJ + + z LaJ + + + + + + ~ SlNnCJ - § z z a: :c 0 + + -J UJ 0 2.s reaction. The two spectra The 1 2.s ground-state group, A comparison spectrum from See page 32 for further discuss ion. used fo r primary Q-value calibration, is slightly off scale at the high-energy end. 12 3 10 c(p, He) B is shown in the insert, near these two levels. sensitive detectors at the focal plane of the spectrograph. shown correspond to the two sets of data collected simultaneously from the two position- 14 3 3 1 Composite He spectra at 20° from the c(p, He) FIGURE 17 II>- Vl I-' + I ] I I I . :n +4' • + i.~ ~t.. + ++ + + ++ + !++++ t g • + 3 PRRTICLE ENERGY CMEV) g ·~· ++\ f <1t\f.+ : ~ ~· +\ + . , ~ + •++ ... ....+ + 128(12.71) 'JI). I 14cfo.3ii.112e I . j I I I rvr.\I · 1 1<,I + I .~. I •• t"lf 128(14.88) 29 I J . x5/(t . ~. I t I A . ~'°-~ • ~ . I + . A: \+ t I ~-1, • 0 + + + ta~ . .p~ 112<: ~,1 I . so~~:V u 100 0 :::> z ~ (f) 150 200 LJij()() 0 :::> z (f) 600 ~ 800 I- . 100 I~ ~ [ it'~ \t. ~ + + + I ,......, I I I I I I + + 113 ~ -J I-' CD 'i ::::: o:i t-'- ':j ( Jl (Jl 1-' ~e e.surements, at 8° in the laboratory. See page 33 for discussion. 3 14 3 1 A typical He spectrum accumulated in a single run, for the c(p, He) 2.s FIGURE 18 (j) ~ I-' F j.gur e i n l 37 • •i • ++ m N .... •• --- 00 Q) J: •••• i + M , i • •• •• .t a. ... .t.•• + + + + + + z z ~ + + u + •• ~ i + z co ........ ...... en •• .t ~ + + + 0 •• \ + •• •• + + + a_ + •• •• - 51 - lll ,..... en ....... w - 8 ie SlNnCJ 3 1 c(p, He) ~ measurements. 14 e.:.E.2.ysis of these data. The uncertainties are The calibration See page 34 for further discussion . The smooth curve was constructed by a least-squares obtained from observations through a ladder mask are shown. s::::a' 1er than the indicated points. ?C~~ts (100 channels = detector length) for the Calibration curve to determine focal-plane position from measured pulse height FIGURE 19 I-' CD (.ll l :59 Figure l 9 ~ z (/') _J 0 .._.. u.J z ~ a: a: . CD .._.. _J a: u z 0 .._.. ~ .._.. z a: u I: CD N --... Q) -...J (") ' ~ z -~a. i3 ...... - u.J I: u.J (/) (/') _J 0 0.... ::::> 0.... -an (I) -- !- 6<W:l ) NrJ I l I SrJd · l!I 140 FIGURE 20 Angular distribution measurements from the experiment. 14 c(p, 3He) 1 2:s The upper half of the figure shows the distribution for the lowest T = 2 state. The lower portion shows the partial distribution measured for the broader group at 14.9-MeV excitation energy. The vertical bars indicate the probable error for the measured cross sections. The horizontal bars show the spectrograph aperture in the center-of-mass system. 35-36 for additional discussion. See pages Figure 20 l4l 150 14c(p,3He)12e ,...., 125 a: en Ex= 12.71 MeV E1 =634 MeV ':1100 al ~ z 0 ....... '15 ~ ,,,u LLJ 50 en en 0 a: 25 u 0 ,...., a:"° en ' al :1. ~ z 30 0 ....... ~ ~20 t Ex=14.86 MeV +++ en en en 010 a: u 0 C.M. ANGLE <DEGREES) 112 FIGURE 2l 3 A comparison of proton spectra at 10° from ( He,p) reactions on c, 160 and 180. 12 See page 43 for discussion. Figure 2l J.43 12c 3000 ..... If) ~ 2000 0 u 1000 + + + 0 (3He, p) reactions 10° 1500 E1=12 MeV If) ..... z :::J 0 1000 u 500 0 180 1000 20i= T=2 ~• .""t•""*"'-"":.·. "·...•..,.•'•..,._..L..-.~: ..r.v·..• '4W! • ~ • •\ NMR FREQUENCY <MHZ) •• + • 144 FIGURE 22 . 3 A comparison of proton spectra at 20° from ( He,p) . 12 reactions on c, 160 and 18O. This shows the spectrum near the predicted location of the first-excited T = 2 state of 2°F; the level identified as T = 2 is indicated. for additional discussion. .. See page 43 145 Figure 2 2 12c If) 30llO to- z :::> c u 2IJllO ICIDO 0 (3He, p) reactions . 20° E1=12 MeV 800 U> to- 160 800 z :::> c u llOO .. o~~.,.,-~~~~~~.:.:~.a:..~.s.a.:.a.i~~.:t...~...l.l.o+*&ti~~:.t.:...~~~.1.-~.E.ll.""""'f..&o...,.,..~~~~~~--I 180 If) 300 t-- z ::> 0 u 200 .... ..... ....·...:..---··..... ~ ........A ......... ... NHR FREQUENCY <MHZ) .. . • ••• . • ••+ .... • .o lowest T = 2 state. 3 2 o( He,p) °F measured by a position- This spectrum shows the region near the 18 See page 43 for additional discussion. sensitive detector, at 5° in the laboratory. A typical spectrum of protons from FIGURE 23 m I-' .;:.. 1 47 LL N ~ Figur e 2 3 • II 1-- z -0 I- • • •• ( J') .. .-..·· 0 CL :.; •• • • •• ... . 5l 8 s1Nnc:i 0 3 for 12-MeV incident He energy. See page 46 for discussion . 2 Angular distribution of the lowest T = 2 state of °F, at 6. 51-MeV excitation, FIGURE 24 m I-' II'> 149 Figure 24 ft0: J~ " en w ..... w l3 UJ 0 ""-/ UJ iS z CI: • 2: • LI § - § - <~S/8TI) I NCI1J3S SSC~J 0 150 FIGURE 25 3 A comparison of' proton spectra at 10° f'rom ( He,p) reactions on cuss ion. 2~. JVJg, 26 Mg, 12 16 C and o. See page 50 f'or dis- 151 NMR FREQUENCY (MHZ> Fignre 2~) 51 for discussion. See page 26 28 3 Mg( He,p) Al measured by a position-sensitive detector, in the vicinity of the lowest T = 2 state, at 10° in the laboratory. A typical proton spectrum from FIGURE 26 u; i' \) !-' 1 53 Figure 26 ~ c: JQ 1 N """' Cf) ~ UJ z z a: ::r: u '-' z 0 ......... -N ""~ + ...... ......... + Cf) 0 Cl- - fil - 8 SlNnrJ:J 0 See page 51 for discussion . 3 28 26 Mg( He,p) Al at 20° in the lab , near the predicted location of the first -excited T = 2 state . Protons from FIGURE 27 (J; .,,.. I-' 155 Ji':le;u re i'_7 ----~--~------~-J • • • >- u z UJ :::> C3 UJ a: LL • • .t • • SlNnCJ a: z~ •• See pages 53-54 for discussion. Angular distribution of the lowest T = 2 state of FIGURE 28 28 Al, at 6.00-MeV excitation. m (Jl I-' 1 57 .... #a. £ I ........ Cf) N w w ffi w a "-JI' w ~ z a: • ~ u ~ - (~S/8TI) - § ~ NCilJ3S SSO~J 0 • 158 FIGURE 29 Angular distribution of the two members of the 28Al doublet near 7.5-MeV excitation, from page 54 for discussion. 26 28 3 Mg( He,p) Al. See 159 Figur e 29 2 '1g(3f.te,p)2.8AI " l&OO ffi Ex=149 MeV ' al :1 ~ I :z 300 a .._. l- u200 UJ . (/') L=1 (/') (/') a 100 a: . u Ex=7.45 MeV :z ~200 tu UJ (/') I I ~100 a a: u C.M. ANGLE CDEGREES) I 160 FIGURE 30 The vector Coulomb energies E(l) as a f'unction of mass c number for T = l and T = 2 multiplets. the coefficient of T z This term is essentially in the quadratic mass equation. 57 for additional discussion. See page l6l Figure 30 8000-----------------------------------..---...... 5000 •T=2 •T=1 ~ a: w t-- 3000 a: 0 t-- Ll w i; 2CXX> en ..... 1000 MASS NUMBER FIGURE 3l 2 The tensor Coulomb energies E( ) as a f'unction of mass c number for T = l and T = 2 multiplets. c is the coefficient of T 2 z This term is %, where in the quadratic mass equation. page 58 for additional discussion. See 163 MASS NUMBER Figure 31 164 FIGURE 32 The isoscalar coefficient from the quadratic mass formula as a function of mass number, for T = O, T = l, and T = 2. See page 59 for discussion. Figure 32 165 qoooo --------~----......------.-----...,.------"'T9'"------ •T=2 • T=1 • T=O ~ lCXXJO l1J ...... a: a: a: u 0 _J en CD Cll..1CXXJO ~ . -20000 MASS NUMBER