A Study of d-He⁴ Elastic Scattering Citation Senhouse, Lionel S. (1964) A Study of d-He⁴ Elastic Scattering. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0X0S-8G45. https://resolver.caltech.edu/CaltechETD:etd-10042002-075353 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated be [...]. Abstract is included in .pdf document. Angular distributions [...] of He[superscript 4](d,d)He[superscript 4] have been obtained, using a gas scattering chamber, for deuteron bombarding energies from 2.935-11.475 MeV at half-MeV intervals. A lithium drafted solid-state detector was used to detect the scattered deuterons and recoil alpha particles. The phase shift and single level analyses performed with the data result in energy levels in Li[superscript 6] at excitation energies of E[subscript x] = 4.57, 4.62 and 7.19 MeV, and have respective J[superscript pi] assignments of 2[superscript +], 1[superscript +] and 1[superscript -]. There is a great degree of confidence in the assignments for the first two levels; however, further analysis is necessary to confirm the assignments to the third level. 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): Tombrello, Thomas A. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1964 Record Number: CaltechETD:etd-10042002-075353 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10042002-075353 DOI: 10.7907/0X0S-8G45 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3889 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 07 Oct 2002 Last Modified: 25 Jan 2024 00:50 Thesis Files Preview PDF (Senhouse_l_1964.pdf) - Final Version See Usage Policy. 6MB Repository Staff Only: item control page

A STUDY OF d-He* ELASTIC SCATTERING Thesis by L. 5S. Senhouse, Jr. In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1964 ACKNOWLEDGMENTS The author is immensely grateful for the inspiration and guidance given him by Dr. Thomas A. Tombrello through the crucial part of this work. The author would, in addition, like to thank Professors Thomas Lauritsen and Charles A. Barnes for having suggested the problem and rendering assistance on many occasions. And finally, the author would like to extend his appreciation to the entire Kellogg staff and personnel and, in particular, to Dr. Peter D. M. Parker and Dr. Phillip A. Seeger for valuable help and sug- gestions throughout the duration of this project. Paul E. Lloyd and Hughes Aircraft Company,in having sup- ported the author by means of fellowships through his course of study, and the Office of Naval Research and the U. S. Atomic Energy Com~ mission,in having supported the project, are all very much deserving of the accolades the author would hereby bestow. ABSTRACT 4 4 Angular distributions (17.97° < 8. m < 156.0°) of He (d,d)He have been obtained, using a gas scattering chamber, for deuteron bombarding energies from 2.935-11.475 MeV at half-MeV intervals. A lithium drifted solid-state detector was used to detect the scattered deuterons and recoil alpha particles. The phase shift and single level analyses performed with the data result in energy levels in Lie at excitation energies of E. = 4.57, 4.62 and 7.19 MeV, and have respec-~ tive J" assignments of 2", 17 and 1. There is a great degree of con- fidence in the assignments for the first two levels; however, further analysis is necessary to confirm the assignments to the third level. PART Il. Til. Iv. VI. TABLE OF CONTENTS INTRODUCTION . GAS SCATTERING CHAMBER Ae B. Assembly and Testing 1. Beam Collimator 2. Faraday Cup 3. Foils. ° 4. Gas Handling and Manometer TITLE Design and Construction 5. Charcoal Trap 6. Detection of Scattered Particles l. Alignment 2. Experimental Checks on Chamber Accuracy a. Faraday Cup Voltages b. Rutherford Scattering c. Multiple Scattering d. Miscellaneous EXPERIMENTAL PROCEDURE RESULTS . . REDUCTION OF EXPERIMENTAL DATA A. B. Co dD. Energy Loss in Foil and Gas Energy of Observed Particles Formulas for Data Reduction Error Analysis ANALYSIS OF DATA Ae Introduction . e 17 21 23 23 23 25 27 a9 29 TABLE OF CONTENTS (2) PART B. Discussion l. Ze TITLE Outline of the Analysis Procedure in the Analysis a. The 2" Level . . The 1 Level . ° The 7.4 MeV Level be Ce C. Summary APPENDICES A. Calculation of Reduction Formulas . B. Sample Calculation for Data Reduction C. Differential Cross-Section as a Function of the Elements of the Collision Matrix . D. Single Level Formulae ° EE. Tensor Polarization Parameters F. Sample Calculation for the Nata Analysis G. Fortran Program for the Data Analysis . REFERENCES TABLES FIGURES PAGE 29 36 40 42, 42 42 46 47 47 52 54 60 63 68 71 74 77 100 Ie INTRODUCTION This study is an extension to higher energies of the investigation of the Lie nucleus by means of the He (4, djHe* reaction. This inves - tigation was made possible through the use of the ONR EN Tandem Van de Graaff constructed by High Voltage Engineering Corporation. The lowest unbound T= 0 level (Fig. 1) with spin, parity and excitation energy assignments of 3°, Ez. = 2.184 MeV was originally investigated by Lauritsen et al. (1953) and further confirmed by Galonsky et al. (1955). The levels studied in this experiment were the T = 0 levels with published assignments of spins, parities and excitation energies of (Ajzenberg-Selove and Lauritsen 1959) + Il. 2; EY = 4-52 MeV 2, 1; E_~ 5.5 MeV 3. 73 E = 7.4 MeV The existence of the first two levels was necessary in the single level parameterization made by Galonsky and McEllistrem (1955a) of data taken up to a center of mass excitation energy of 4.55 MeV. In that the data were taken essentially at the resonance energy of the first level, one would feel confident of level assignments of the first level. The energies of the first and third levels are obtained from the Li'(He?, ajLi? reaction {Allen et al. 1960). The spectrum of this reaction had rather salient peaks at the energies corresponding to the first and third levels and, therefore, one would expect these ener gy assignments to be reliable. The assignments of the second level, however, remain to be confirmed, because this state does not appear strongly in the Li (He?, a)Li? experiment. The state was obtained from the alpha particle spectrum of the experiment after subtracting close-lying prominent levels (especially E = 5.35 Mev}. The state was also inferred from the He (a, d)He* experiment (Galonsky et al. 1955) by the interpretation of data removed from the excitation energy of the level by an MeV. There is, on the other hand, considerable evidence of a theoretical nature for the first two states. The states have resulted from inter- mediate coupling models by Inglis (1952, 1953) and Balashov (1959). The states have also resulted from the cluster model of Wildermuth et al: (1961) and the three-particle model of Wackman and Austern (1962). The experimental investigation of these three levels was com- plicated by the existence of a three-particle reaction channel, which can proceed in one or more of three possible ways (Lauritsen and Ajzenberg-Selove 1962), l. Het +d > He +p Het t+ pin Qn 7 732182 MeV 4 .5 4 2. He td-Li° +n->-He tptn Qn 7 74 192 MeV 4 3. He +d- Het +ptn Qa -2.225 MeV There is reason to believe, from an analysis of the neutron yield (Lefevre etal. 1962}, that the first two predominate, while the last probably involves the Is state for p + n (Rybakov etal. 1961), which would make the reaction isotopic spin forbidden. Further proof for the predominance of the first two and the absence of the third is found in the threshold for He (a, pn)He* being 0.4 MeV above the calculated threshold in the center of mass (Henkel et al. 1955) for the third reaction. The value for the total reaction cross-section is 0.3 + 0,1 barn at an incident energy of 10.3 MeV for the deuterons (Allred et al. 1951). This dissertation presents the reduction of the elastic scattering data as angular distributions and excitation curves after first discuss- ing the design, construction, assembly and testing of the gas scattering chamber with which the data were obtained. Subsequently, the experi- mental cross-sections are compared with those of a phase shift analysis. The program used for this analysis appears in its entirety in Appendix G and is presented schematically in Figure 12. The pro- gram, in addition to performing a phase shift analysis, calculates tensor polarization parameters and values for the total reaction cross- section. Finally, a single level parameterization of the phase shifts yields single level parameters for the three levels investigated. In addition to the intrinsic worth of an extension lo higher ener gies of the investigation of the Li? nucleus by means of the He “(d, d)He* reaction, this study performs other functions. It corroborates the angular distribution data taken on a cyclotron by Stewart et al. (1962) but disagrees with the 8.0 MeV data of Burge et al. (1952). Also, the analysis of the data enables one to investigate the usefulness ofa phase shift analysis involving a large number of parameters at high energies. It might, in addition, be hoped that one day, in the not too distant future, a more comprehensive theory of the nucleus will evolve, imparting more meaning to this and the many other efforts that are continually being undertaken. fl. GAS SCATTERING CHAMBER A. Design and Construction Gas scattering chambers are found in the literature in two basic types: those that use foils to separate the target gas from the high vacuum in the beam tube and in the Faraday cup, and those that use differential pumping to maintain the pressure gradient between the high vacuum in the beam tube and the target gas, retaining the foil between the gas and the Faraday cup. The sizes of gas scattering chambers also vary; one of the largest in the literature has an inside diameter of 36 inches (Worthington et al. 1953), and one of the smallest used in a scattering experiment is about 3 inches with fixed angles of observation (Huntoon et al. 1940). The size of the scattering chamber is determined mainly by the detectors and by consideration of wall scattering. In this experiment, the decision was made to use solid state detectors, which are rather compact (7/8''O.D. x 1 mm. thick) in comparison to proportional counters (Worthington 1953). Further- more, it was possible through the use of antiscattering slits in the detector collimators to shield from wall scattering, permitting the use of a chamber of only 11$'' inside diameter and yet retaining a detector with a continuously variable angular position. ‘The gas scattering chamber constructed for the present experi- ment is shown in Figure 2. A steel cylinder is provided with an entrance beam collimator (3) and a beam collector (16} located on a diameter. Located within the chamber are two detector assemblies (9) and { 10), one fixed and the other (10) movable in angle. The entire chamber is filled with the scattering gas, and the scattering volume is defined by the beam collimator and the slit system attached to the detectors. Vacuum seals are effected by foils (2) and (12) at the entrance to the collimator and the collector respectively. The collector volume is connected by vacuum plumbing (not shown) to the main vacuum system of the beam tube. The main body of the chamber is a welded fabrication made from 12" steel tubular stock with a bottom plate 1" thick. Electrical con- nections are brought into the vacuum through kovar glass-to-metal seals and O-rings are used as mechanical vacuum seals. The chamber has an entrance port in which the beam collimator is mounted, an exit port on which a Faraday cup is mounted, and a hole in the base for the rotating hub on which the adjustable platform is mounted. In addition, there are ports for evacuation, gas handling and impurity trapping. The chamber was rough drilled, welded, and then stress relieved before the finish cuts on a milling machine. In this way, it was possible to hold the orthogonality of the center line of the entrance port to that of the detector platform to within + 0.05°. Under normal operating conditions, the chamber is separated from high vacuum by a 2500 A or 5000 A nickel entrance foil before the beam collimator and a 10,000 A nickel exit foil before the Faraday cup. This enables one to maintain a gas pressure in the chamber of about 20 cm.-oil, while ~6 the high vacuum side is on the order of 10 mm. of Hg. i.e Bear Collimator In order to provide an accurately collimated beam with a minimum of scattering, the beam collimator ((3) in Fig. 2) is provided with a series of disc apertures, made of .020"' thick tantalum. The direction~ ‘ality of the collimator was insured by drilling and boring the inner dimensions for the apertures and then turning the outer chamber mating diameters concentric with the inside diameters. The apertures were first drilled, then mounted on small arbors for + 0.001" concentric turning on the outer diameter. Six apertures are provided; two ((4) of Fig. 2) have 0.052" diameter holes and serve to define the beam; the remaining four have 0.0635" diameter holes and serve to stop scattered particles. 2. Faraday Cup The beam collector cup ({12-16) in Fig. 2) is provided with a $"' hole, sealed with a nickel foil. Graphical checks of the geometry by the method of Dickinson and Dodder (1953) indicated that less than 0.2% of the beam should be scattered to larger diameter than $"', at the highest helium pressure used. Charge collection is accomplished by an insulated cup ({16) in Fig. 2), which is provided with an electron suppression ring ((15) in Fig. 2). The collecting cup is maintained at +67% V. with respect to the grounded chamber, and the suppressor at ~300 V. Although the geometrical arrangement was by no means ideal, checks of a mock-up in an electrolytic bath indicated that with sufficient potential on the suppressor the system should be effective in stopping secondary electrons. Later checks with an alpha particle beam indicated that the suppression was not complete, and in the final runs, an Alnico magnet {(13) in Fig. 2) was added. With this arrangement, it is believed that the charge collection was accurate to better than 0.2%. 3. Foils As previously noted, foils were used to isolate the scattering chamber from the main vacuum system. As compared with a differential pumping channel, foils introduce somewhat larger energy loss and straggling in the beam, but in view of the large breadth of the levels observed, this difference was of no importance. In order to permit easy replacement of the entrance foils, they were mounted in inserts in a modified plug shutoff valve, as shown in Figure 2 (1,2). The quarter-inch diameter of the orifice in the entrance-foil mount was determined by the diameter of the focused beam at that point. The half-inch diameter orifice in the exit~foil mount was decided on by considering the diameter of the multiply scattered beam. The gas pressure of about 20 cm.-oil in the chamber was decided on by a consideration of energy loss in the gas and rate of the scattered-particle yield. With these constraints, the choice of 2500 A or 5000 A nickel entrance foils and 10,000 A nickel exit foils proved suitable. Froma strength consideration, even thinner foils could have been mounted at the entrance. However, thinner foils obtained from the supplier (Chromium Corporation of America) had an abnormal propensity for holes. The mounts for the foils were deburred of sharp edges and were so designed that the pressure in the chamber would push the foils against their mounts. The reason for this is to reduce the possibility of insufficient adherence of the bonding agent, which in this case is solder. Solder was chosen because of its reliability and its ease of use as a bond. The foils were mounted by heating the mounts on a hot plate, tinning with the solder and affixing the foils. The 5000 A and 10,000 A foils were self-supporting and con- sequently could be inspected for holes before mounting. The 2500 A foil was not self-supporting, but was obtained from the supplier on an opague copper backing. This backing was removed with a solution of equal parts trichloroacetic acid, ammonium hydroxide and water. Upon removal of the backing, a flashlight was used to illuminate the back of the foil, while the passage of light through possible holes was viewed from the front. Invariably, this test was sufficient. 10 4. Gas Handling and Manometer The gas handling consists of a network of quarter-inch copper tubing linking one or two gas bottles with the chamber proper through quarter-inch Hoke needle valves. In addition, there is a compound gauge that monitors the pressure in the gas line. The manometer is connected between the chamber proper and the high vacuum through 3/8" copper tubing. The fluid used is n-butyl sebacate (Russell ct al. 1956}, because of ita low vapor pressure. Its specific gravity was measured to be 0.927 + 0.003. The manometer, being in excess of one meter, was long enough for the pressures used. A cathetometer was used for accuracy in making the pressure readings. Although the cathetometer can be read to within 0.001 cm., the process used in setting it up reduced the accuracy to approximately + 0.03 cm. 5. Charcoal Trap In order to remove significant contaminants, e.g. oxygen and nitrogen, a 2'' diameter charcoal trap is mounted in the bottom of the chamber. At liquid nitrogen temperatures, this trap sufficed to keep the total contaminant yield below 1% of the scattered or recoil-particle yield at all angles. Because the trap absorbs deuterium, it was not cooled when deuterium was in the chamber, 6. Detection of Scattered Particles Two detectors were used: one was capable of motion for angular distributions and the second was stationary for monitoring the beam as 11 a check on the consistency of the current measurement. The moving detector was a 1 mm. thick lithium drifted solid state detector with a 5/8" diameter active area and overall diameter of 7/8". The lithium drifted detector had a resolution of 1.5% for 8.778 MeV pp*!4 alpha particles, using the electronic setup described in ''Experimental Procedure."' The lithium drifted detector is an extension of the p-n diffused junction (Pell 1960). ‘he drawback with a normal p-n detector is the need for high reverse bias voltages in order to acquire a thick enough depletion depth for complete energy collection from fast particles. The lithium drifted detector, on the other hand, does not need a large reverse bias for a thick depletion depth, for the detector--by virtue of boing a drifted p-ien junction--is inherently deep and only needs bias for ion pair collection. The fabrication. of the p-i-n junction is accomplished by first depositing lithium on a slab of low resistivity (100 G-cm.)p-«type silicon. The lithium is then diffused and subsequently heated with reverse vias on the silicon. What happens at this point is believed to be a compensa - tion for the p acceptor type silicon by a deposition of the lithium donor atoms interstitially. The result is an intrinsic region, the depth of which depends on the voltage, temperature and time elapsed in drifting. ” The detector was made by J. R. Yvon Cusson from a method developed by M. Emery Nordberg, Jre and J. R. Yvon Cusson. 12 Itis only possible to compensate the material, at most, and not possible to change the n-type region to p or the p-type region to n in that reverse fields would set up a reverse drift, if over-compensation were imminent. On completion of the drift process, the back of the detector is carefully etched and used as the exposed surface for particle entry, using a gold evaporation for electrical contact. The monitor was an 8500 Q-cm. surface barrier detector, fab- ricated by the author. Although the effective thickness was somewhat less (6-7 MeV protons) than that of the movable detector, the resolution was quite adequate for the purpose. Both detectors were provided with collimators, consisting of rectangular openings milled in 1/16'' steel. The shapes of the openings for the moving detector were chosen in such a way as to simplify the data reduction (Appendix A) and to minimize second order corrections. The dimensions, as determined finally by measurements on a compar~ ator, were Front slit: 0.08145" + 0.4% wide Rear slit: 0.04088 + 0.5% inch“ area; 0.0825" wide h = 2.3325" + 0.005% (Figure 10) Ro = 4.79" +0. 1% (Figure 10) The monitor detector, with its collimating assembly, was affixed to the chamber cover plate ((9) in Fig. 2), at an angle of about 30° to the incident beam. The movable detector was mounted on a carefully 13 machined hub ((18) in Fig. 2) which could be rotated from outside the chamber. A vernier protractor permitted angle settings accurate to 6' of arc. A sliding post in the center of the hub establishes the geometrical center of the rotating system. B. Assembly and Testing 1. Alignment The philosophy of the assembling of the chamber was to be as careful and precise as possible, then check by repeating some well~ known gas scattering experiments. The first step in the assembly was to mount the moving detector and collimator on the adjustable platform in the proper orientation. This was accomplished by making the sub- assembly in a mill, using a dial indicator to ascertain that the slit was vertical. The counter mount was then assembled in the chamber, setting the distance from a vertical flat on the center post to the front slit to 2.46". After assembly, the chamber was aligned with respect to the optical axis of the beam tube. This optical axis had been defined by the zero position of two pairs of orthogonal adjustable slits in the beam tube and a bench mark on the wall of the target room. A Brunson telescope was set on the upstream end of the beam tube collinear with the optical axis; then the chamber was positioned while viewing the defining apertures of the beam collimator to insure collinearity. The 14 moving counter collimator was then lined up with the detector removed by sighting the pattern made by the slits of the collimator, the point of the center post and the apertures of the beam collimator. 2. Experimental Checks on the Accuracy of the Chamber In a first attempt at designing and constructing any apparatus with a measure of complexity, there arc inevitably flaws in the resulting product. To enumerate the extent and degree of that imperfection in this tome would not only enlarge it significantly, but would be to a great extent embarrassing in the recounting. Suffice it to say, there were numerous preliminary runs on the chamber that pointed up numerous imperfections, which were corrected. The presentation here will dis- cuss the chamber as a finished product and the final tests that exude an air of respectability in their relating. a. Faraday Cup Voltages The Faraday cup was run positively biased and the suppressor or guard ring was run negatively biased. Electrostatic suppression proved sufficient in the case of deuterons as incident particles, but it was necessary to introduce magnetic suppression in the case of alpha particles below 6 MeV (Section II-B -b). The necessary value for the voltage on the suppressor to stop electrons from the exit foil and the necessary value for the voltage on the cup to prevent electrons from leaving the cup were obtained by connecting the monitor to a scaler 15 and noting the variation in counts for a prescribed total charge. It can be seen from Figure 3 that the charge collection is independent of the voltage on the cup and becomes independent of that on the suppressor at about -80 volts. Typical running values were -300 volts on the suppressor and +67%# volts on the cup with the additional aid of the magnetic suppression. b. Rutherford Scattering The chamber was filled with tank argon of 99.5% purity, and the elastic scattering of 4.5-MeV alpha particles determined for Oo om. =x 30-160°. The first experiments, carried out with only electro- static electron suppression in the beam collector, showed consistent deviations from Rutherford scattering of about 15%. After installation of a magnetic suppressor \(Section ll-A~Z), data were taken at 30° and 60°. The ratio of the cross-section to Rutherford was 1.03 at 30° and 1.06 at 60°, in keeping with the accuracy of the experiment. ce Multiple Scattering The chamber was checked for multiple scattering at 12° labor- 4 4 . 4 4 atory angle for He (a,a)He at 4 MeV and for He. (d,d)He” at 3 Mev, by varying the pressure of the chamber from about 6 cm. -oil to about 25 cm.-oil. The results are shown in Figures 4 and 5, and indicate that multiple scattering corrections to our data would in all cases have been less than 1.0%. 16 d. Miscelianeous In addition to the above tests, there were several others checking minor features of the chamber. These included verification of the assumption that the thermometer in the chamber measured the temper- ature in the scattering volume. If the temperature throughout the chamber is not uniform, one would expect to see variations in monitor counts when the beam current is changed. This test was run with alpha particle beams from 0.075 ua. to 0.025 ua. and the counts for a given integrated charge showed no systematic variation, validating our assumption. It was also found necessary during our runs to eliminate back- ground arising mainly from neulron reactions in the silicon detectors. These neutrons were caused primarily by the stripping of deuterons on the first slit and the entrance foil. In order to minimize the effect of these neutrons, a heavily shielded slit was placed before the foil to stop excess beam and the forward half of the scattering chamber was filled with steel, the most effective neutron shielding for a small volume. 17 Ill. EXPERIMENTAL PROCEDURE A schematic drawing of the experimental setup is given in Figure 4+ 6. The beam of accelerated particles~-in this instance deuterons, He or He*** with energies from 3-11.5 MeV--is provided by the tandem accelerator. After energy analysis in the 90° deflecting magnet, focus - ing and collimation, the beam passes through the entrance foil, the entrance collector, through the chamber and into the Faraday cup. ‘Within the chamber, particles scattered in a volume of the gas defined by the cross-section of the beam and the geometry of the counter col- limators are detected in the movable and fixed solid-state counters. | The signals from the counters are amplified by Tennelec model #100 fast (~0.1 us.) charge-sensitive cascade preamplifier (Fig. 7) and Hamner model N328 non-overloading, double delay line, clipped linear amplifier and finally into a Radiation Instrument Development Laboratory 400 channel analyzer. Pulses from the fixed {monitor} counter are fed into an Eldorado model SC-700 decimal scaler. The beam collected in the Faraday cup was integrated with an Eldorado model CI-110 current integrator, which is calibrated with a test current supplied by a standard cell that is measured witha galvanometer and a 0.9985 + 0.1% MQ resistor (see Fig. 8). The clock used for timing the total charge is accurate to within + 0.1%, 18 Since the integrator was calibrated at the same current as that of the beam, the error due to leakage currents, internal to the integrator, was minimized. The typical procedure in collecting experimental data is as follaws. The chamber is opened to the roughing pump of the main pumping station, while the charcoal trap is heated. During this time, bias voltages are applied to the detectors and they are checked out on the oscilloscope to see whether they are excessively noisy. In addition, the current integrator is set, and the total charge determined by running a known current, calculated from a galvanomter reading and a resistor, against a prescribed time to accumulate the desired amount of charge. After the charcoal trap has been heated sufficiently, the diffusion pump is connected to the system, which is pumped until a pressure of 107? mm. Hg. is reached. At this time, the center part of the chamber is closed off and filled with gas. The charcoal trap is then cooled to liquid nitrogen temperatures and kept cold throughout the experiment unless deuterium is the gas in the chamber, in which case the trap is left at room temperature. Both temperature and pressure are measured at frequent intervals during the run. The chamber is now ready for beam. The beam alignment is accomplished by a trial and error procedure with the aid of electrostatic and magnetic steering throughout the tandem system, focusing with quadrupole magnets on a viewing quartz interposed just bcfore the entrance 19 foil. The quartz is then removed and the beam maximized in the Faraday cup by again adjusting the steering and focusing of the beam. The runs are now made, with a running check on the pressure, temperature, clock time and live time of the analyzer. Changes in counter angle are accomplished manually. UWpon completion of the runs, the integrator is again checked and is found to remain stable to within the accuracy of the measurements. The preponderance of data are those obtained from He*(d, d)He’, although for a few of the low energy runs, data from H“(a,0)H* are used. A typical pulse-height spectrum obtained withdeuteron bombardment is shown in Figure 11] (E, = 5 MeV, Bab = 35°). In this spectrum may be seen the deuteron peak at channel number 61 and the alpha particle peak at channel number 41. Background counts are superimposed on these peaks, starting off with a relatively high value at low channel numbers and rapidly falling to zero at higher channel numbers. ‘The contaminant spectrum is above channel 64, but does not appear in the figure, since there are only about 10 counts in the peak. At higher energies, the peaks both decrease in size with the number of counts in the deuteron peak falling bclow the number of counts in the alpha peak. At larger laboratory angles, the peaks appear at lower channel numbers and finally are absorbed in the background. The information from the alpha peak can no longer be resolved from the background at @ = 45° lab and the deuterons can no longer be resolved from the background at 90°. 20 The spectra for the monitor counter are essentially the same as those of the moving counter with the exception that at 8-9 MeV the deuteron was energetic enough to penetrate the depletion region in the counter, resulting in pulses being dropped to lower channels, correspond- ing to the fraction of the total energy lost in the depletion region. 2 2 It was only at energies of E, < 4.5 MeV that a few H (v,¢)H d runs were made, because of the reduction in background when alpha particles are used in place of deuterons as the incident projectiles. The benefit from the reduction in background is outweighed, however, by the limitation in center of mass energy and angle (Figs. 9A and 9B} when using H (a. o)H*. Asaresult, the angular distributions from this reaction were taken only at Bf? x 2.935, 3.441, 3.946 and 4.450 MeV and angles from 50° to 130° at 10° intervals in the center of mass (Figs. 18 through 21). 21 IV. RESULTS The main body of data comprises angular distributions at various bombarding energies, Ea from 3 to 11.5 MeV and excitation functions at 18, 55, 90, 130 and 156 degrees (Figs. 18-42). These data have compared with those of Galonsky et al. (1955) (Fig. 39) and Stewart et al. (1962) (Fig. 41). There is a tendency for both the previous works to be somewhat high with respect to the present study, but the agreement on the whole is fair. This work is in definite disagreement with Burge et al. (1952) (Fig. 29). At this juncture, clarification of the data presentation is made. The points on the Figures 18 through 42 are experimental with relative error bars and the solid lines are the phase shifts analysis fit to the data. In those instances where an analysis was not made, dashed lines are drawn through the experimental points for clarity. It should be noted that Figures 39 and 40, which appear at first to be identical, are in fact different in emphasis. The data on Figure 39 was taken expressly for the excitation curve, whereas Figure 40 is an excitation curve compiled from angular distribution data. The data of Figure 40 extend up to 11.5 MeV and suggest a negative curvature in the vicinity of 9 MeV that is not readily apparent in that of Figure 39. 22 The levels of interest here are the T = 0 levels in Li’, which are the only ones that can be reached with d-He* elastic scattering. These are in particular the E = 4.52, 5.5 and 7.4 MeV levels (Ajzenberg- Selove and Lauritsen 1959). It was found (Fig. 38) that 4.52 MeV is the only prominent level; the 5.5-MeV level manifests itself as a slight dip in the cross-section at 90° (Figs. 39 and 40) in keeping with its 7 assignment (Galonsky and McEllistrem 1955a); the 7.4-MeV_ level manifests itself both in the variation in the character of angular dis- tributions from about 6.5 MeV (Figs. 23 through 36) and ina negative curvature in the 90° excitation curve (Fig. 40), suggesting (but not determining) the presence of a negative parity state (Galonsky and McEllistrem 1955a). This argument will be pursued further in the section ''Analysis of Data." 23 V. REDUCTION OF EXPERIMENTAL DATA A. Energy Loss in Foil and Gas In order to determine the actual energy of the incident particle, it was necessary to make a correction for the energy loss in the entrance nickel foil and in the gas between the entrance foil and the scattering volume. This was done through the use of the stopping cross-section for protons, ey? plotted by Whaling (1958) for HH’, Ni and He? as the target materials. The values of the stopping cross- sections for alpha particles and deuterons were computed with eS (E) = fe, (E/3-97)and eq (E,) =e, (e,/2). where E and E. are the respective energies of alpha particles and deuteron particles. The densities used for nickel, helium and deuterium were 8.75 gms./cc., 0.17847 gms./liter and 0.17976 gms. /liter respectively at standard temperature and pressure conditions. The thickness of foils stated by the supplier was assumed correct. The distance from the entrance foil to the center of the chamber is 8.719". The results are tabulated in Tables I through III. Be Energy of Observed Particles In order to differentiate the alpha particle from the deuterium particle in the spectrum of the angular distribution, itis necessary to know something about the kinematics of the reaction. This is easily 24 obtained by using Equations 2.3 through 2.7 on pages 412ff. of Evans (1955), which gives E/E =4M.M cos“(8 ),.4M,+M ) (1) fe) 1 2 r“lab I 2 r and - . “1. E/E, = {Miu + m,)"} {coe8 ant [om /M,)°- sin“(9),.,/} where the 'nuclear disintegration energy'' is equal to zero for elastic scattering and minor changes in notation are made giving the following: Ez is the energy of the recoil particle E is the energy of the scattered particle E is the energy of the incident particle is the mass of the incident particle M is the mass of the target particle (2) is the angle of scattering for the scattered 1 . * particle in the laboratory {e } is the angle of recoil for the struck r‘lab . . particle in the laboratory It is seen that Equation 1 is always single-valued, but Equation 2 is single-valued when M, > M, and double-valued when M, < M, in 2 keeping with the argument on page 412 of Evans (1955), that the square root of the ratio of the enerpies is real and positive. The graphs for 4 d on He? and He on d are Figures 9A and 9B respectively. 25 Cc. Formulas for Data Reduction The formulas. together with their approximations for the differ~ ential cross -section as a function of yield and experimental parameters, are derived in Appendix A. The errors due to the approxirnations, statistics and the inaccurate knowledge of the experirnental parameters are discussed in the following section. Here we present rnerely the formulas for the differential cross-section in the center of mass due to the yield of the incident particle (do/dQ) = (Y _/NnG) sin(®.) ab (ao, ,/49 ) (3) cm and that due to the yield of the recoil particle (4) {do/dQ) me (vy _/4NnG) tan (8 diab where Ge 2bA/R bh and Y. is the yield of scattered particles Y is the yield of recoil particles r N is the number of beam particles during the counting period n is the number of target particles per cubic centimeter A is the area of the second collimator defining slit (Fig. 2) 2b is the width of the first collimator defining slit (Fig. 10) R is the distance from the center of the chamber to the ° back of the second defining slit (Fig. 10) 26 h is the distance from the front of the first to the back of the second defining slit (Fig. 10) The kinematics for the incident particle in a two-body reaction is discussed in Marion and Ginzbarg (1949), and the conversion from /dQ 4 were lab angle to c.m. 1 d th lid angl tio, dQ ab angle to c.m. angle an e solid angle ratio lab em obtained from their tables for the scattered particle, using x = M,/M,,; where M, is the mass of the incident particle and M, is the mass of the struck particle. These tables were calculated from _ 2 _ 2 dQ /daQ F cos(8_- 8, wn) ein 8, ./ sin 0 am (5) “lab ecm A discussion of these kinematics can also be obtained in Evans * (1955), where the following formulas are derived for lab angle toc.m. angle conversion, corte ). iy = (M,/M,)csc(6_) . m o a + cot{@ 5) (6) Ce Me for the scattered particle from which Equation 5 is obtained, and (@ dep 2 Or - (8). 2 7) lab ~ for the recoil particle, which gives together with dQ /4%, = sin dé __/ sin 8 ab d8, ob (8) the obvious result ac tap! oem = tan @ ant sin8 (9) ale . Equations 6 and 7 are the same as Equations 23 and 24 on p. 834 of the cited text with changes in notation. 27 where (9 slom is the angle of scattering in the center of mass ) m is the angle of scattering in the center of mass determined from the recoil particle in the laboratory system D. Error Analysis The analysis of the probable error in this experiment was divided into two parts. One, which is angle-independent and which we choose to call the systematic error, is motivated by Equations 3 and 4 and is tabulated in Table IV. The important factors are the accuracy of the geometric factor, G, the accuracy of charge collection manifested in the beam rate, N, the error in the pressure and temperature manifested in the particle density, n. In addition, the error due to the approxima- tion in the differential cross-section reduction formulas is calculated. The total systematic error due to these factors combines to give an rms value for the probable systematic error of 1%. The relative error arises from several sources. The error in estimating the background (taken to be 1/4 of the background, N/4 (Fig. 11)) contributes. Also, the statistics of the total number of counts in the peak, including the background, M+N, and separately the statistics of the background, N, contribute. These contributions give a relative error of (M + 2N + N’/ 16). In addition, there are other factors contributing to the relative error; among these are the errors 28 acising from the uncertainty in the alignment of the chamber (+ 6°) and the inaccuracy in setting a given protractor angle (also + 6'). These inaccuracies are reflected in the sin(@ ), | (49, ,/dQ__) and slab cm lab tan(@_) terms of Equations 3 and 4. Tabulations of this effect are lab in Tables V and VI. A further effect of the two sources of the + 6! uncertainty is an uncertainty as to the true value of the measured yield. This effect of approximately 1.5% predominates at 17.97° and 156° in the center of mass, at which angies this effect is included. At the same angles (17.97° and 156°), an additional effect is of significance, that of the error resulting from a high dead-time correction. This error was found to be on the order of 2% for the large dead-time correction of 30%. The procedure used for determining the dead-time correction was that of comparing the live-time meter of the 400 channel analyzer with a pulser connected to the analyzer and a scaler. There are sources of error that were estimated to be negligible. The check on the dead-time correction shows that itis in error by less than 1% at angles other than 17.97° and 156°. Furthermore, the error in yield due to uncertainty in angle is also less than 1% at angles other than 17.97° and 156°. Also, extrapolation of the contaminant yield to forward angles, where it could not be resolved from the desired particle yield, indicates errors of less than 1% and might be considered to be accounted for in the background subtraction. Finally, the mean é 2 charge of 2 used for H“(a,0)H is not in error by more than 0.6%. 29 VI. ANALYSIS OF DATA A. Introduction 1. Outline of the Analysis Here the choice of approach in analyzing the elastic scattering data will be recapitulated. With the rather sparse evidence for any level other than the 2" (Section IV), it was decided not to use a single level approximation to begin with, but to use a parametrization in terms of phase shifts with real and imaginary parts and tensor coupling parameters. The derivation of the scattering cross-section formula ak Ss 2 Jal 44 (B\*+ sin“e{|c|*+ |p |*} + sin’9 { |z|*} is outlined in Appendix C and is equation C.10, where do aa is the differential scattering cross-section in the center BY of mass system k is the wave number for the reduced mass A through E are the transition amplitudes from channel spin state m_ to channel spin state m_,, where 8 s 2,0 _ 2,40 _ 2 k Cs at 41 =k Caa).4 -1 |A| 2,do 2 aa )0,0 = |B| KC" 2 2 2 5 (do/49), 44 =(k"/sin’6) {do/49) 1g = (c|"/2 sin 9 30 (x"/sin"e)(do/a"), ,, = (7 /sin“e)(de/an), _, = |D|7/2 0,+1 (k°/ sin" (do /an), 12 (K"/sin“eacsan) | = |e |*/2 These parameters are given in terms of the scattering matrix U, in equations C.11 through C.15. Each of the above amplitudes conserves Jv, and consists of a part that conserves @ and a part that allows # to change by t 2. In addition, amplitudes A and B have a Rutherford scattering term because m is conserved for these elements. Even though one does not obtain the quantitative values for the widths or excitation energies of the levels in using a phase shift analysis, one does get an indication of the resonating terms, in that they tend to go positive instead of continuing the negative trend due to potential or hard sphere scattering. Furthermore, once the phase shift parameters are obtained, single level parameters can be calcu- lated on the assumption of an interaction distance. This procedure is outlined in Appendix D and the level parameters tabulated in Tables VII through IX. Once the phase shift analysis was decided upon, it was necessary to determine the parameters to use. Galonsky and McEllistrem (1955a) used s-wave and d-wave in their single-level analysis of the data up to EA = 4,62 MeV. Itis therefore obvious that s through d waves should be included in the analysis. This resulted in three parameters for the 3 a d-wave (6, 1 5 ’ a) and one paramctcr for the s-wave {85 ). In 3] view of the departure of p-wave from the hard sphere values in the analysis of Galonsky and McEllistrem, it was decided to use three . 2 1 0 was . parameters for the pewave{d, ’ by , 8) ). In addition, there is a possibility of coupling between levels of the same Jn, so a coupling . . 1 1 constant ¢ was introduced for coupling between 65 and é. (Gammel et al., 1960). The above real parameters do not allow for the three particle reaction (see Introduction). In order to allow for the three particle reaction, it was decided to make all the aforementioned parameters complex, At this juncture, one might consider the possible configurations for the E = 7.4 MeV level, which appears to have a negative parity . 4 4 (p. 22) and must be T = 0 if it is to be observed in the He -(d, d)He a reaction. The possibility of additional p states was discounted at this energy, in that Inglis (1952, 1953) predicted, through the use of inlecmediale coupling, that the next pe level was al E = 14.0 MeV. Also, p configurations result in positive parity states. From the former configuration, there are two possible T = 0 states: one, the space anti-symmetric singlet state that can be made 1 with 8 ; two, the space symmetric triplet state that can be made with 0 5°, § 7 and aT - The lpld configuration allows several additional possibilities. Among the space anti-symmetric states, there are those - And formed by 85° and oar in addition to the one furmed by 6 1 among the space symmetric states, there are the ones formed by 4 3 2 1 6, » 6, ; 8, » § 3 bo and 6, ; in addition to the one formed by > a) om bo and 6. Due to an error on the part of the author in determining the space symmetries, the greater fraction of these possibilities was not con- sidered. And what is even more unfortunate, the more likely possibility CP 210! received only a cursory examination. All of the phase shifts, however, are included in the program with the exception of 6 In 3 e was . 1 1 . addition, a coupling parameter between by and 6, and a coupling parameter between 6 1 and 8. were included. The possibilities of varying the real and/or imaginary parts of the parameters were included, which resuited in a total of nine cases, and are self-explanatory in the copy of the program (Appendix G). Before continuing, it would be advisable to look at the form of the scattering matrix. Lane and Thomas (1958) show that when the scattering matrix represents the totality of events in the universe, it must be unitary and symmetric. The unitarity insures conservation of particles with a total probability of one, and the symmetry insures the time reversibility of Schridinger's cquation. In the case of d-He elastic scattering, where the reaction channels are not treated explicity in our formulation, it is necessary to account for a loss in particles by making the phase shifts imaginary. 33 In the case where there is not a coupling between channels of different £ values, the scattering is diagonal in the phase shift, é, ’ but when there is a coupling between two £ values, the diagonal matrix must be rotated by a 2x 2 unitary matrix (Blatt and Biedenharn 1952). This results in a coupling parameter or rotation angle. ¢. in two space. A 2x 2 scattering matrix without a loss of particles and without tensor coupling can be expressed by et ey Us ol &5 (10) 0 > @e where 84 and 85 are real. When coupling between channels 1 and 2 is considered, U must then be operated on with the rotation matrix / cosé ’ sine T = (11) -sine >» cose where & is real. Then, vid vis. zis vid. cos“ce a ein“e e , $sin(zeYe to e ay -1 Ul'=T U qT vib, wis 2 zié 2 wid, & Sin(ze je -e ), sin“c e + cos ¢ce (12) It is readily seen that within the constraints on the scattering matrix, three additional parameters can be used by considering those 34 alrcady used as being complex instcad of real. Furthermore, if there were no coupling, it is obvious that complex phase shifts give an amplitude reduction or a loss in particles. The object of the program (Figure 12) is to calculate, using a ‘starting set of phase shifts and coupling parameters, a scattering cross-section for each angle and comparing it with the experimental cross-section. On this comparison, a necessary change in the initial parameters is derived, and if this change is significant, the initial parameters are changed, and a new set of cross-sections is calculated. This continues, until the changes become insignificant. Our motivation in calculating the change in the starting parameters is a modification of Newton's method: I o ) =(o +) (3 aé.pas, tr0° 13 ( ex! ( calc! ‘Ly ( Scale! ? A i (13) n n , n i=l where I is the number of parameters N is the number of angles in an angular distribution (o } is the ni? experimental differential cross-section ex’ on (oie) is the n calculated differential cross-section n é, is the ih parameter . .th Ab; is the change in thei parameter 35 One has, in general, N pieces of data for N angles at a given energy, and therefore can form matrices with N rows. Letting z =(c ) -{o ) , one obtains column matrices with N rows for n ex calc a z and I rows for A$,. The matrix A for (ac alc! 08? has I columns Cc and N rows. In order to calculate A 6,9 one would normaily proceed by multiplying A and z by the transpose of A, then multiplying both sides by the inverse of Ata, This results in Ab = Tay (ate) (14) This was tried, but did not converge. The argument is that necessary precision is lost on taking the inverse, even though a double precision matrix inversion routine was used. It was decided that the mathematical grounds for the method used to obtain convergence were incidental, provided a method of convergence could be found. In this spirit, a gross simplification was made by considering all of 2 to be due to the 5. under consideration, which resulted in ad A&,=N )) z [A . (15) 1 n ni n=l after averaging the A 6i due to each n over the number of pieces of data, N. The procedure resulting from equation 15 did not work any better than that of equation 14. It was decided that the problem might have been over-simplified. In order to better approximate the method of equation 14 and yet avoid a matrix inversion, it was decided to 36 ignore the off-diagonal elements of Ata and use the equation N N A 6. Fd ), Zo Any / ) Any (16) n=l n=l which proved successful. In the program, once a satisfactory set of values is obtained for the parameters, the iteration stops and the tensor polarization param- eters, which are the invariant parameters of double scattering (Appendix E), and the total reaction cross-section are calculated. The formula used for the total reaction cross-section was £+1 2 a c= (n/*) ; { (2441) -L(2043)/31| B - [(2e+1)/3] | uF 2 - [(2#-1)/3]| uf , \°- [(2e+3)/3] | Uh eee |". [(2e-1)/311y } (17) in keeping with Blatt and Weisskopf (1952a), p. 431. 2. Procedure in the Analysis The phase shift analysis, in its original form, was coded for the Burroughs 220 in machine language. The differences between the 220 program and that finally coded in Fortran for the IBM 7090 (Appendix G) was that the former did not include tensor coupling and imaginary phase shifts for particle loss in the reaction channels. In addition, the 37 decision-making logic used for the 220 program was much simpler in that the author was present to make logical decisions throughout the iterative procedure. The Burroughs 220 program was initially checked by using the phase shifts Galonsky and McEllistrem (1955a) report at the 1.07 MeV level in the program and confirming the cross-sections they report. Unfortunately, these data were all at back-scattering angles and did not point up a program error in the Rutherford part of the scattering cross-section. This error was revealed on comparing the 220 program with the 7090 program, using the same input data. In addition, the accuracy of the corrected 220 program was checked by hand calculations for simple input data and further compared with the 7090 program. This did not give a check on ¢ or on the imaginary parts of the phase shifts. However, all errors that would show up in internal inconsistencies (e.g., the existence of a total reaction cross-section with all real variable parameters) were found. Also, visual checking and rechecking of the program was done. In the region from 3 to 4.5 MeV, the object of the program was to verify the results of Galonsky and McElistrem (1955a). This was done, mainly, by trying to eliminate the need for the _ resonance, in that it was not salient in the He*(d, d)He* excitation curves. This procedure did not work. Asa result of the phase shift analysis in the 2 1 3 to 5 MeV region, anomalous behavior of 6 and &, was necessary. 2 2 38 As further corroboration. of the work of Galonsky and McFEllistrem {1955a), it was not possible to continue the p-wave according to hard sphere scattering, but instead there was a trend for all the p-waves to stay above -20° (Figure 15). The real part of ¢ was included in the analysis and the value for it ranged from 14° at 2.935 MeV to 0. 8° at 4.955 MeV (Figure 13) for the best fits. This behavior seems unrealistic, especially in view of the prediction that ¢ is insignificant in this energy region (Gammel et al. 1960). An explanation of this ‘behavior might be obtained from considering the small number of data points used in this region with somewhat large error bars. The analysis from 5 to 7 MeV proved to be particularly challeng- ing. The reasons for this are many. First, there is a three particle reaction that manifests itself through one or more of eight possible channels, if we limit ourselves to £ = 2 waves asa maximum. Further - more, the real part of ¢ was predicted to be of significance in this energy range (Gammel et al. 1960). Moreover, there is the change of characteristic in the angular distributions that suggests a resonance, which can come about through one or more of seven possibilities, if the arbitrary restriction £ = 2 maximum is imposed. If f-waves are also considered, six additional parameters occure In view of the large number of possibilities, it was necessary to exercise judgment and concentrate on the possibilities that were more likely. 39 In spite of the complications in this energy region, it was possible to continue the real parts of the s-wave and the d-waves with reasonable confidence {Figures 13 and 14), The 64 Parameter is in good agree~ 3 ment with that predicted by Gammel et al. (1960) and the 65 parameter is not unreasonable. The resonating behavior of 5” and 5 will be discussed further below. An attempt was made in this energy region to eliminate the 7.4 MeV level with no real success. Consequently, a decision had to be made about the pewaves. The departure from hard sphere behavior 0 1 continued in this region with fy and 84 being especially conspicuous, There is reason (see "Outline of the Analysis") to expect a resonance through & » but the attempt was made to give & resonant behavior. 1 1 This will be discussed further below. On investigating the consequences of an imaginary part for e, it was shown that this should be small or zero, depending on the imagin- ary parts of 55. and 8), in order to prevent a creation of particles (Tombrello 1963). The real parts of ¢ proved to be slightly negative (-2° ) after averaging over the best fits in this energy region. In view of the fact that the scatter about the average was in excess of 2° , it would not be unreasonable to use zero as the value of ¢ in this energy region. The three particle reaction tended to perturb the imaginary 2 parts of 65° and 55 ; and upon calculating the excitation curve for 40 the total reaction cross-section, using the average values of the parameters obtained in the phase shift analysis, a rather convincing curve resulted (Figure 17). This curve, on extrapolation of the lower end, is in rather good agreement with the threshold of the reaction (Henkel et al- 1955), and on extrapolation at the high-energy end, is in good agreement with the reported total cross-section of 0.3 + 0.1 barn (Allred et al. 1951), In that the total reaction cross-section was not used as a constraint in the analysis, this convincing result would indicate that the reaction cross-section is fairly well determined by \ the scattering data. The tensor polarization parameters resulting from the analysis are in reasonable agreement with those of Haeberli (1963) and Seiler et al. (1962) and are tabulated in Tables X through XXIII. + a. The 2 Level The values for 4. are in good agreement with those of Galonsky and McEllistrem (1955a) up to 4.5 MeV (Eigure 14). The continuation of this parameter through the transition region (5 to 7 MeV) presented some difficulty, in that there was a tendency for it to level off at any one of several values starting at 120°. The continuation chosen was the one consistent with a small value of ¢ in both the real and imaginary 2 parts. Once the values of the 6, phase shift were obtained, a reli- ability check was made on them by introducing reasonable variations 4) 2 in the pewaves. The values for & remained fairly insensitive. 2 From the values of § om a single level approximation was made, 2 : using a value of 4.8 fermis for the interaction radius, which gave the best agreement in the resonant region (8 = 90°) of several radii tried, including that of 3.5 fermis used by Galonsky and McEllistrem (1955a). The reason the interaction radius of 3.5 fermis is acceptable for Galonsky and McEllistrem's (1955a) analysis,but unacceptable for the analysis of the present experiment, stems from the variation between the 84. of the for mer experiment and that of the latter (Figure 14) in } the region of resonance. Once the resonant phase shifts, 8, are cal- culated from the level assignments of each of the aforementioned | experiments, the resonant phase shifts of both of the experiments arc exceptionally compatible. On the other hand, the values for the resonant phase shifts of the present experiment , obtained from the single level approximation, differ quite markedly (15°-20°) from the values obtained from the phase shift analysis in the neighborhood of Ey =9 MeV. This deviation would imply that it is impossible to make a single level approximation to the scattering data throughout the resonating region, holding the interaction radius constant. The assign- ments resulting from the single approximation are tabulated in Table VII. 42 + be The 1 Level The continuation of the _ phase shift through the transition region was easier and the values for this parameter less sensitive than those of the remaining parameters. A single level parameterization of this level was also made with an interaction radius of 4.8 fermis, the results of which are tabulated in Table VIII. ce The 7.4 MeV Level The behavior of oo chosen as the parameter of the resonance . is extremely poor. The ''mean'" curve was drawn through points with a great deal of scatter-(Figure 15). The fit of the angular distributions calculated from these ' 'mean'"' values for the parameters to the experi- mental points leaves something to be desired at all energies, with the possible exception of 7.968 MeV. These deviations from good fits might be attributed to the p-wave inconsistencies. In spite of the lack of confidence in the 1 assignment for the 7.4 MeV level, a single level parameterization was made, using an interaction radius of 3.5 fermis, which gives the correct excitation energy. The results of the parameterization are tabulated in Table IX. B. Discussion The phase shift analy sis corroborated Galonsky and McEllistrem (1955a) in the 3 to 4.5 MeV range. Furthermore, there was a measure 43 of success in the fitting through the transition region (5 to 7 MeV). The successful aspects were the determining of assignments for the 2" and if levels, a convincing prediction for the shape of the total cross-section for the three particle reaction and reasonable values for tensor polarization parameters. The one great disappointment is the uncertainty of the level assignments for the 7.4 MeV level evidence in the erratic behavior of the p-waves, Suggestions will be made for further study in this area after dispensing with the question of unique- ness of the phase shift analysis obtained from an elastic scattering experiment. The problem of the uniqueness of a phase shift analysis fit is embodied in the number of parameters needed tu describe the elastic scattering process of spin 1 particles on spin zero particles (Phillips 1960; Puzikov 1957). Equation C.10, although written with five param- eters, each of which depends on all 2 values, can be reduced to four (Gammel et al. 1960). These four parameters, in contrast to the case of one parameter for spin zero on spin zero particles, cannot be uniquely determined from angular distributions. The results of angular distributions present one constraint. In order to provide for the three additional constraints, it is necessary to perform double scattering experiments. The equation of import is (Stapp 1957) 44 1'(8', @') = 1(8') [1 + A(8',o') + B(e',') cos op! + C(@!, ep") cos 2") (18) where I'{6'~) is the differential cross-section after the second scattering 1,'(@ ‘'y is the differential cross-section for an unpolarized beam and is measured in the elastic scattering experiment A, BandC are parameters depending on the center of mass energy and the scattering angles 8' and! are the angles of the second scattering using the plane formed by the first scattering as the x'z' planc and the direction of the beam after the scattering as the z' axis On examining equation 18, it is seen that by judicious choices ina double scattering experiment of the angles 9! for a given angle 9', the values for A, B and C can be determined. The non-uniqueness of the phase shift analysis is probably not as bleak as it appears to be at first sight, in that there is a great deal of physical intuition in picking a starting set of values and insuring the continuity of the fit throughout the energy range considered. The problem arices if there are two solutions existing in the same area. 45 This was not investigated at length, but would seem improbable for those parameters that were insensitive and quite possible for those that were not. In view of the large number of possible parameters in the fitting of the data and the lack of uniqueness in any given fit, it would seem advisable at this point to press for further theoretical study in order to determine more probable choices from the overwhelming possibil - ities. This study could center about the likelihood of low-lying odd parity states and their expected space symmetries and J values. In the case of triplet states, it would be informative to have a treatment of the splitting and whether the split has L +S or |L -S( as the lowest energy state. In the realm of a theoretical treatment, one might transcend the immediate problem and investigate the possible break- down at high energies of the phase shift analysis and/or the single level approximation. In addition to the above, and in keeping with the non-uniqueness of the elastic scattering fit, it would be advisable to incorporate the data of a double scattering experiment into the constraints on a phase shift analy sis. In lieu of striking out for new horizons, the more pedestrian approach may be taken in a concerted effort to understand the p-wave behavior in the elastic scattering. The 3p state (see "Outline of the Analysis'') is most probable, and an attempt should be made to force a § ° resonance in conjunction with 6,1 and 8} . The § fa phase 46 shift,in conjunction with the 4 phase shift, is also suggested in the 1 1 space symmetric triplel resulling froma lpld configuration, and it might prove fruitful to force a resonance using both these phase shifts at the same time. The possibility remains, in spite of the convincing evidence to the contrary, that the three particle reaction is not going through the correct channels as yet. It might be expected, in that the 2 and the + 1 levels are resonating in the transition region, that the particle loss . . 2 1. should go through the imaginary parts of 64 and 8, instead of 8, 2 and b5 As a final alternative, f-waves under more intensive study might present the solutions for dilemmas that still remain. CG. Summary In summation, the phase shifts obtained are presented in Figures 13 through 16 after having drawn a ''mean" curve through the calculated points. The fits to the angular distributions are presented in Figures 18 through 31 for the ''mean" data. In addition, the assignments for the three levels investigated are tabulated in Tables VII through IX. The tensor polarization parameters are tabulated in Tables X through XXIII, and the total reaction cross-section is plotted in Figure 17. 47 APPENDIX A Calculation of Reduction Formulas In Figure 10, the origin of coordinates is located at the center of the rear aperture of the collimator. The z-axis is the centerline of the collimating slits, intersecting the beam line atz=R_. The beam lies in the xz-plane, making an angle 8, with the z-axis. The edge of the front slit is the line x = b, z =h. | We direct our attention to a bundle of particles leaving an element of the beam line dB, at distance B along the beam line from its inter - section with the z-axis. The number of particles arriving at an area element dS in the xy-plane is -2 dY = Nn dB. of(8) dS cosyR (A. 1a) where x is the angle between the normal to dS and the ray to dB. In terms of the angle 9 between the beam line B and the ray between dB and ds, dB =Rd0/sin9 fA. 1b) and dY =Nn o{8) cos x dS d@ /R sing {A. 1c) where a(9) = (do/d2) lab {A.2) Furthermore, cos X¥ ={R - B cos 6 /R (A. 3) rR = (Ro -B cos8.)* + y- +(x -B sin 8) (A. 4) 48 ao, . 2 a R’ sin’§ =(R_ sin @ -xcos@}) +y (A.5) fe) re) fe) On squaring the terms in{A.4) and solving for B, one obtains B={R_ cos 0 txsind )t{(R cose +x in)” fe) o oo o fe) o 2, 2, 2 12,48 -Rooty tx -R)) (A. 6) Collecting terms B#R _ cos$8 +x sin68 +R cos 8 (A. 7) fay oO on“ evolves, which gives on making the appropriate substitution of Equation A.5 B#R_ cos 6 +xsin 6 +(cos 6/sin aie sin@ -xcos6 ) re) ) o--* Oo o re) 55 ry bF Ca.) It is obvious from Figure 11 that B < RG cos 6 +x sin ey so the minus sign in Equation A.8is retained. It can also be seen from Figure 10 that the limits on B are given by (x - B sin 8 MRS - B cos 6) = (x + b)/h (A.9) Now, let& #0 -90 ,c =cos0,S # sin 9. Then by a Taylor o 86° o Oo ° series expansion, one obtains 2 2, 3 cot: @=c /S -&/S “-c_ & /S {A. 10) oO o 9) o . On making this substitution into (A. 8) and expanding 49 ole 2 2 2 RS => + } = Ss { 1e- s +4 R { ( 00 * co) 7 Ro Oo * c/R, ce) 2 Ge co! 080 + Hy /R 8)” - Bee IRS} 2 S - 4 . eR Si -xc tay (RS, (A.11) : . 3 3 one obtains upon neglecting x andy terms 2 2 B=x/S ~-y~c /2R S “+e(R_/S -xc /S )-2'R c/s * ° fo) rome) o oO o O oo 0 (A. 12) up to second order in §. Using 2 -1 2 2,01 2 ins = a RS - + = + é - — /2 (R“sin 8) sin 6[ oo xc) y ] (Ss, Ec § / Ss / 2 2 x .. 13 [RSj-xe) ty ] (4.. 13) 2 2.-1 ~2 7 s - + = 5 1+ 2 5 AR, oF c,) y J (Ry ) C * co/R, fo) 2 2 + 3(x c /RS.) - (y/R 8.) 7 {A.. 14) and (A.12), one obtains, continuing the approximation 2. ~1 (R_ -Bcos8® )/R sind=(RS) (ltxc /RS o fe) ome) , 0.8008 (O 2 2 -2 2 2 + S) "+4 - 2)(R_S - 2 4.15 (xc /RS)" + Ble, - 2UR{S,) yy - 8/2) (4-15) Given (A. 9), one has the limits on B and can determine the limits on &. This is accomplished by using {A. 12) and temporarily 2 neglecting § terms and solving for © with the appropriate limits on B. One obtains ~ 2 a 2 2-2 ea -(x Fb)/ht+y c(/2R | S + Fb) c/s, (A. 16) However, it can be seen from (A. 12) that if 50 2 2 eF=-(xFb)/hty c /2R | 5, (A.17) then the extra term in (A. 16) is of the order e. Expanding 2 o(8) = o(@ |) + Boe ) + de o'(8) (A.18) and substituting (A.3), (A. 4), and{A.5) into (A.1), one gets Nnds fo(9) cos x (R sin a) d, = ~] ¢ a Nnds(R S.)” f { of8,) + 8o'(8.) + BE om(8 Dt (A.. 19) 2 2 -2.2 2 x {1 + xc /RS | + {x c/R 8) + alc, -2)(R 5) y -& /2} dé Using (A. 17) for the limits on &, integrating over area A and continuing the approximation, one obtains - 20-1 2,2 2. 24, 2m Na(R S$.) Aal8.)° bh {ite x, ISR te) (8 °R tHe *-2) -2,2 RS) OD, | 2 pion af"), J2n+ Col@ Ylol8 )It-x, /h+ ely), /28 Ro” (A. 20) - cs(e"), /S RBI M8 )/0(8))(0° /6h@+ (x") 2n’)} -l 2 for the yield, where x =A [ x dS, x av av -1 =A | x dS, and 2 -1 2 Y ay 74 fy ds. ; 2 Noting from Equation A. 20 that if b jhe << 1, one can use 1 «1 - (RS )7 Ao (@ ) 2bh A.2l Nn(R 5.) o {@ ) A ) except for small values of So an attempt is made to design the slit *Equation A.20 differs from Breit (1939) in that the x4) 2he term should have a minus sign instead of the plus Breit et al. (1939) give. 51 system so that (A.21) is valid at all angles. In considering the additional terms in (A. 20), itis seen that terms without Sy in the denominator can be neglected at small angles and, in addition, xy terms are zero if the slitis symmetric. Furthermore, it can be assumed that Ruther- ford scattering is the primary contributor to the scattering at small angles and has the cross section o(9.)=c/e * (A. 22) fo) fe) : where C is a constant. Then for a rectangular slit with half width b, one obtains b -i = 2 x5 [ xox = {A. 23) fo) b 2 2 1 2 b xo =a | x dx = (A. 24) fe) 2 (2 £ Yay" TE (4.25) where £ is the length of the rear slit. Substituting Equations A. 22 through A.25 in A.20, equating (A. 20) to {A.21), and neglecting b-/ 6h”, one obtains (2 /2)° = 2(sn°)"* (n“ + 4R oh + 20 Ro‘) (A. 26) Also, this equation has been derived with a high degree of accuracy by Silverstein (1959). The high order corrections resulting from Silver - stein's derivation are insignificant in comparison to the accuracy of this experiment. 5a APPENDIX B Sample Calculation for Data Reduction Run #23 on February 13, 1962 will be used in the sample calcula- tion. The nominal energy of this run is 5 Mev. and the spectrum is plotted in Figure 11. We use Equations 3 through 9 and first calculate G = 2b,A/R hb = .08145x 1040875 x2. 54/4.79 x2. 3325 = 7.587x10 om. N = Vxtx60/Rxl. 602x1071%= £04901 x 5.023 x 60/.9985 x 10°x 1.402 x 107 *? = 9.234 i0'? particles 18 n = 6.5839 x 10° x P/T P/T =.07911 The value of Y . is obtained by subtracting N' from M' + N' and cor- recting with the ratio of clock time over live time Y. = M! x 337/333 = (13756 - 360) x 337/333 and the value of Y is obtained by subtracting N from M + N and cor- recting with the ratio of clock time over live time Y= M x 337/333 = (16028 - 1200) x 337/333 The value for 8 is found for 6 = 35° and M_/M. = 0.5 in Marion cm lab 1 2 and Ginzbarg (1949) and is equal to 51.67°. The value of dQ [dQ on lab for this angle is .5122 and sin Sab = 0.57358. Combining these quantities according to Equation 3, we get (do/do) = .1093 barns/ster. 53 at 51.67° c.m. For the rccoil particle, we are able to find oom by subtracting 2 x 9 from 180°, which gives 8 om = 110°. Using tané@ lab = .70021, we get (do/da) = 0.0719 barns/ster. for 110° c.m. lab The value for the corrected energy of 4.955 MeV is found in Table III for 5 Mev deuterons on 5000 A nickel foil and 23.6 cm oil. The relative error on the differential cross-section in the case of the scattered particle is obtained by noting that the vector sum of vive + 2N' +N "7 16)= 1.11% and [sin(®, + J2x%. 1)(do, /dQ ), -Sin 6 b cm lab ] (do, /42 pV +f/2x. lab’ lab cme, sin 6, , (d2 lab /d2 ) lab cm = .306x/2% give an rms error of + 1.0% for o. when rounded off. In the case of the 2 recoil particle, we geLM + ecNtN /16 = 2.21% and [tan(9 +/2x.1 = tan Sap /tan 8 = /2x .408% lab lab when added vectorially and rounded off, give an rms error of + 2.0% for oo: The error in dead time correction and the error in the yield due to an uncertainty in the laboratory angle is not significant at this data point. On this date, the cnergy was known to be better than 3/1000, which gives an rms error of + .015 MeV. 54 APPENDIX C Differential Cross-Section as a Punction of the Dlements of the Collision Matrix The derivation of the general equation can be found in many dissertations (Sachs 1953). The analysis is made possible by the short- range nature of the nuclear force. This enables one to solve the Schrodinger equation without a detailed knowledge of the form of the nuclear potential, in that the solutions are outside the range of the potential. Because of this, itis only necessary to include the long~ range Coulomb potential explicitly. The asymptotic solution of the wave function is indeterminate within phase factors in its exponential representation. These phase factors are called phase shifts and their values are caused by the nuclear force that shifts their values from what they would be in the absence of a nuclear potential (Hvans 1955). In the collision matrix representation of the differential cross- section, it is possible to set up the wave function for the scattering process by considering the asymptotic solutions to the Schrédinger wave equation, both the incoming part due to the incident beam and an outgoing part, proportional to the elements of the collision matrix, due to scat- tering. Since the ultimate goal is to represent the elements of the col- lision matrix with a finite number of phase shifts in an open form, it is necessary to extract the Rutherford part, which is dependent on all £ values. Itis then possible, after proper normalization, to identify the differential cross-section with the square of the absolute value of the outgoing amplitude. The result is (do/d0) = (nik, “| x ¢7 (20+1)Fexp(dila,+ a, )CU-1) vo!) £ if R ; tis | (c.1) Jn oo which is the same as Equation 10.11 of Sachs (1953) with a= a(n, -T) 3 -2i7) | 9° andR_ =t"e p_- o a £ -1, 7 a, = 2) tan (+) the Coulomb phase shift with a = 0 s=] | 1 = az! oe /hy | z is the atomic number of the incident particle is the atomic number of the target particle e is the charge of the electron h is Planck's constant divided by 2r k is the wave number in the center of mass £ is the orbital quantum number U is the collision matrix I is the unit matrix ' ym (c) is the spherical harmonic of the outgoing wave given t by Galonsky and McEllistrem(1955a) 56 Ae m! 4 (m'! m' ! tlimtly! a vPro) =(-yF Ge tte Dp A ele (2 '+ lm'|)! (m' ) _|m'| i my d P (cos 8) x sin {c.2) . d (cos 9) mT where £, 2 £ P(x) =| 1 d ea} (c.3) 2g! a is the identity of the particle, including spin orientation - s denotes £2 and mas wellaso primes refer to outgoing particles unprimes refer to incoming particles The Rutherford amplitude of the incident wave Ro =-k4| csc“(8/2) exp (i7] In cse“(8/2)) In Equation C.1, the U's are in an (S,2, mg:m,) representation. In order to finally evaluate the elements of U, it is necessary to make the transformation to an (S,4, J,m,) representation (Blatt and Bieden- harn 1952). This is accomplished by considering m m m J. tSeI 4 s | = 3Im_,2,S8/m_.,m.,£, C.4 ¥ 5 #3 )) (Im,.£,8/m, 14,8) Yy x (c.4) + = ‘which is the coupling of the eigenfunctions of the spin operator and the orbital angular momentum operators to form a wave function that is an eigenfunction of the total angular momentum which is a conserved 57. quantity in the elastic scattering interaction. Equation C.4is sug- gested by Sachs (1953) in his Appendix 3 and differs from that of +S- Blatt and Biedenharn in the phase factor if SJ where (J,m_,2 S/m, , maf ,S) is the representation for Clebsch-Gordan co- J efficients defined by Condon and Shortley (1935). This modification is necessary for the preservation of invariance under time reversal. * Upon the conversion of (S',#',m mit m a Jul ¥, gem, '|U|S,£,m J gus! % a factor of i acises where £-2' —~ 0, + 2 in the 4 wo! case of d-He elastic scattering. This, when multiplied by ra f 0,+4 originally in Equation C.1, gives i ’— , which in all cases equals 1. 5° ™,) to < *Invariance under time reversal was neglected by Galonsky and McEllistrem (1952a), which means the off-diagonal elements of their collision matrix have the wrong signs. This oversight was of no con- sequence in their case, in that they assumed they could neglect the off-diagonal elements and would also be inconsequential in this dis- sertation, since the sign could be absorbed in the phase of the coupling parameter (Appendix A). The invariance under time reversal concept is then academic, but yet interesting. This is discussed in Baldin et al. (1961). §, 21, the upshot of which is in using the time reversal operation in quantum mechanics, we should get effects similar to that in classical mechanics. For instance, if we have a dynamic equation that has proceeded accord- ing to Newton's postulates, we should be able to retrace our steps by replacing quantities such as position x by x and velocity v by -v. In the case of magnetic field, it can be resolved, with a little thought, that the field would have to be inverted in the time reversed equation for a charged particle to retrace its steps. It can be shown (Wigner 1932) that time reversal in quantum). | mechanics is equivalent to an ogeration on the operator O of KV OVE, where V is a unitary matrix, V its hermetian adjoint, and K is the complex conjugate operation. Itis also shown in Baldin et al. (1961) that an operation on the eigenfunction of VK Yo is equivalent to operating on O. Furthermore, it can be shown that for rotational operators such as the angular momentum operator L_, analogous to the situation in classical mechanics, the operation K TL VK = “EO . 58 4 Upon specializing the equation to d-He elastic scattering and defining the parameters A through E in terms of (do/dQ) rm 2 one obtains ss 2 2 2 k (do/d9) ay =k (do/d%) = |A| {c.5) 2 k (do /40) , v= IB|? (c.6) 2 (k°/sin“e)ao/40), , 0 =(k /sin“6)(do/a0)_, 0” \c|*/2 (C.7) 2 2 2 (k"/sin"@)(do/da), ., = (k / sin“o)(do/ 4), \D|*/2 (C. 8) (7) sin “e)(do/42), + (k°/sin*8)(do/ do) = |B\*/2 (c.9) -l,+1 In order to obtain the differential cross-section for an unpolarized beam, we must sum the above equations over the outgoing channels and average over the incoming ones, which gives 3k°(do/ do) = z\a|? + js |? + sin“e { \cl@ + 1 |*} + sin“e { ||} (C.10) where ziorg Lic o+1 g A =Rtx(e* 4j { ® P +2 20+ R+Ple /4i )4 e , LU Ut (ee 1) U, 1 d-l . aay 42 a gt] + (g-1 ~ +e* + +2))* (g Uy 2(2£+1)]+e Py gt le 1)(¢+2)) Uy p42 LIOp . -1 tee 4 7 Pe (ae-iFu, f ; i (c.11) 59 - ; - (20+1 B =R#F (ce /2i) 4 e PlU+yU,y +L Uy y - (2441) din Big eat By ail £-2 -e “pp [le+er2)]* U a.et2 7° Poa Bgl z C.12 +tee-]? u, 92 f (c. 12) zie, gio, atl c-Fle £103) {te * P,/a(eri)] Celet2) U, {241 U, 1: 2 bel, 42, Bot] ~@°-NU | jhe Pog Cer ter2yl" Oy p42 gi 4 re Py ig EME-DE Uy 97 J | {c-12) Bid, gia, gio g+1 £-1 gt2 =. ' - + P ' D = Fle *12i) {¢ PU, , -Uy, Ite 442 1: 1. gid 1 Beth. £-2,, , By 1 x [(et1)/e+2)]* Uy - [ee -1)] tat (c. 14) i: Ls pid RIM» E=2 {e *ravezid{te * Peery] rete, et (eth) u, ; £ ~ > g-1 Bid 4 1 o+l ~ j 2 "W +(et1)U, 5 Jtfe /T+12+2)1°) Py" U, iyo 1: Biv . . ~Z a: -l +(e . /Cae{e-1)]*)P, io" U f (C. 15) £-2 £,4-2 where U, ag! is the element of the scattering matrix, conserving J and going from an incident channel £ to a final channel £'. 60 APPENDIX D Single Level Formulas In this appendix, the formulas used in the single level approxima- tion to each of the resonances will be presented. The formulas, when used for those £-values through which there are reactions in addition to elastic scattering, are rather poor approximations to the data of this dissertation, in that the reaction channels are neglected in the formulas. The single level analysis, which is valid for resonances removed from others of the same Jr, results from the following (Galonsky and McEllistrem 1955a). g f = 24 + = ¢D. Uy yg 7 exel itp, +8 )1,(5 = 4) {D. 1) gt Zig, = = Tr {2ip°—") U £41 =e £¢ A, £te + Af exp ).{J=2+1) (D. 2) d d a * 2 y they fhe Hoyt ey 2 TL Th esa) g+2 “og, 242° © r +1 +1 x 2i sin eg’ exp(ip’= ), (J = 441) (D. 3) tan a7 = (T, /2)/(E, -E),{T, aT for J =2£ = + for J=£t1 D.4 ye Tate 4) (©.4) 2 2 Tr /2 2 (k A D.5 yl Z yg WA, (0-5) BE =E +A D.6 Roh OX (D. 6) ee! A, = - = [g, +2) for J 22 (D.7) 61 2 VY, 92 ¥ _ Arf N,kt 2 = - oe + - + $+2 =pt A, = lg, £] = [ey 42 4+2 | for J =2£+1 (p.8) d(In A_) 2 d(In (ka)) tan Pp) = - (F/G), (D. 10) where 2 yf is the reduced width of the ) level leading to the decay of the Lie state into a deuteron and alpha particle £ is the relative orbital angular momentum My f is proportional to the probability for decay aa is an inhibiting Coulomb and centrifugal A £ 1 harrier factor = F a tG 2 £ £ FY is the regular Coulomb function Gy is the irregular Coulomb function k is the wave number Py is the hard-sphere scattering phase shift Bp is the resonance scattering phase shift I ; U, g! is an element of the scattering matrix a is the interaction radius in the scattering From Equations D.4, D.5, D.6, and D.7, one obtains 62 2 ne ka - [g, +4 + +z 7 1=E (D.11) A, tan B r a The above equation was used to fit the phase shifts obtained from the J J phase shift analysis with the assumption that 6 = ®, + By » The £ parameters gy and ?) were computed, using Equations D.9 and D.10, from a knowledge of the regular and irregular Coulomb functions. The limitations of Equation D.11 are worth repeating. In using (D.11}, we are in effect ignoring the possibility of tensor force coupling or reactions through channels other than elastic scattering, both of which are only approximately true in this dissertation. 63 APPENDIX E Tensor Polarization Parameters The tensor invariants of the double scattering process, which are tabulated in Tables X through XXIII,will be discussed and presented in this appendix. The outgoing wave in elastic scattering of a polarized beam is in a mixed state and can be represented as a linear combination Ss of the pure states x (Pondrom 1959), m S _V\ i s m ! ° s where ¥(m .) is the outgoing wave Xe \ is the eigenfunction of the Schrédinger wave s equation with the operator S. m is the eigenvalue of the spin projection of the polarized beam in the initial state m _' is the projection on the z axis of the spin s (m |M| xm") is the transition amplitude from state m to state m ' 4, In that the dimension of the spin space is three for d-He elastic scat- tering, it is possible to characterize the density matrix in terms of a linear combination of nine (3°) irreducible Hermetian operators TM (Lakin 1955), which must be chosen to satisfy the commutators (Racah 1942) 64 as i , = + 1)1* a, ry +3 J) ave [(J + MMI +M + 1) Ty, Mtl (EH. 2) and [J ve *“MT,, (E.3) where + M =(- 4 Tou (-1) Ty oM (&. 4) These operators transform like spherical harmonics with their normal- ization and orthogonality being chosen as . t © Tom Fo rt = 30 ayn gM! (&.5) Asa result, one obtains a tensor of rank zero (the identity matrix), three tensors of rank one, and five tensors of rank two: =. : Ti15 eV/3 (8 +18) (&. 6) tr, =-(2)%s (E.7) 10 2 Zz ° T.. = #/3 {8 +18 \* (BE. 8) 2a * Pa y =-4 i +s (S_ +i . TS) 5/3 [(s_ +i s.) sts) (s. is) {E.9) and 4 2 To5 = (2)° (3S ~ - 2) (E. 10) These are expressions for five of the matrices in terms of spin oper- ator S, with the ones for negative M being determined from (E.4). The Equations E.6 through E.10 can be verified with (E.2), (E.3), and (E.5), realizing that the T's commute with the orbital part of the J's. 65 The representations of the spin operators can be obtained by defining ] 0 0 S = 0 0 0 Zz 0 0 -1 0 1 0 S$ 2% 1 0 1 x f2 0 l 0 and finally, with the use of the commutator, [S_. SJ =iS y 0 -i 0 1 S == i 0 -i y J2 0 +i 0 By considering Equation E.1, the expectation value for the T's of a beam that is initially in the uw th substate can be written as + (alT|n) == (BM [nd (B8/ T= “|My? |y) B v + 7 (B(M |w> (u|M|y> <p/Tly > in the Dirac bra and ket notation. For a beam that is initially unpolar- ized, itis now necessary to average over the values of 66 1 {ret Ow} OM = sa OE ite) . 1 1 1 too, =, (asa = (B8|M |w) cu lelyy (8|Tly) which gives (TS = oo x op. T (E.11) where t _ 1 > oM MoM. (em. 12) the By element of the density matrix and Ss, = 1, the initial spin state. The value for M is a -g e M = f b -f e g a where az=A beB g=sin98C//2 f = sin 8D/ /2 e = sin’ 0 E/ /2 with A through E being defined in Appendix C, which results in tro = 80 lal? + alo]? + lel? + e124 ela (E. 13) 67 i(T,)> = (ve (Tre) (bf) + Im(ge’) + Im(ag)] (E. (7,5) =(/y / Trp) [2 Re (ae) - \¢|77 (EB. (T,)) = -({ Jz /Trp) [Re (bf) + Re(e’g) - Re(ag’)] (E. (75) = CE step) Cla]? + fel? + fel? 2 fal®= [bla Xe. (Thy? =0 (E. 14) 15) 16) 17) 18) 68 APPENDIX F Sample Calculation for the Data Analysis In this appendix, a set of starting values for the phase shift analysis will be presented, using a program identical with that of Appendix G with one exception. The program in Appendix G forces the imaginary parts of the coupling parameter negative, whereas the pro- gram that is used for this sample forced this parameter positive. The angular distribution of interest is that of 7.968 MeV. Some degree of analysis has already taken place, and it is now of interest to find the effect of starting the program with the real parts of the phase shifts equal to that of the best fit previously obtained, | but with the departure from the previous fit by using Im 85 < Im 84%. In accord with the above, the values for the real parts are: 1 o 2 o by = -15.5 1 = 25.8° 8 5.8 0 ° 5 = -15.5 3 ° 8, = 171.25 2 ° 8, = 136.19 6.) = 111.09° 2 —~ cf -1.28° a iY 69 and the values for the imaginary parts are: by =0.0° 8, = 0,0° 5 = 3.0° 8° =0.0° 85 = 16.0° oe =22.0° 8 = 5.0° € = 0,0° Case 2 of the program is chosen which docs not affect the real parts, but does perturb the imaginary parts, resulting in: By «= 040° 6° = 0.81" 8) = 5.95" 7 = 1.06° 8 = 29.95° 8° = 10.90° 65 = 0.0° 70 ‘Results for tensor polarization parameters and the total reaction cross- sections are also obtained, but will not be presented here. Upon plotting the phase shift parameters at this and other energies, then fairing the data, one obtains the results tabulated in captions of the Figures 13 through 17. One also obtains the tensor polarization parameters tabulated in Table XXI. It is now possible to calculate the level parameters of an energy level, e.g. the 2 level. Itis decided to use 4.8 fermis for the inter- action radius and it is now necessary to calculate 9,, Ay g,» and k (Appendix D) using a Burroughs 220 program coded for this computation. It is then possible, using the values for the 8 phase shifts, to cal- culate 8, = § 2 2 27 Pos The valuc for the excitation energy is that at which 8, =90°. This energy is 4.57 MeV. Next, Equation D.11 is used for EA = 3.0 to 6.0 MeV, anda series of seven simultaneous equations are formed by pairing each of these obtained with the one obtained at the resonating energy. This results in a total of seven values for 2 Yn a ,; ranging from 0.738 MeV to 1.083 MeV. The average of 0.981 a MeV is used. On calculating the Wigner limit from né Ma 3 2 one obtains 71 APPENDIX G Fortran Program tor Data Analysis This program is described in the section entitled ''Outline of the Analysis" and is presented in an abridged version in Figure 12. In this treatment, a copy of the coding is presented along with a description of the input and the output format. A definition of the input quantities is as follows: NMAX is the total number of differential cross-sections in the angular distribution IMAX is the number of starting parameters used, where the real and the imaginary parts of a phase shift are two parameters KW is the control word, indicating the case desired; this can be 1 through 9 with the function of each case being described in the program comments WGT is a weighting factor, that is set equal to one T is the laboratory energy of the incident deuteron in MeV SIGEX(N) is the experimental cross-section of the nh Piece of data ANGLE(N) is the center of mass angle of the nt? piece of data LMAX or MAXL is the maximum value of £ included in the iteration: this is coupled with the case chosen to some extent 72 t DELTA(I) is the I h parameter in the program and is equal to: I Re &4 T=l1 Re 5," E= 2 Res? I =3 1 0 Re 6, 1T=#4 3 Re 6, T=5 2 Re 6, T=6 R 31 l=7 e 2 = R = ee) I=8 3 Re 6, I=9 2 = 10 Re é, I Ree, I- lil and to the imaginary parts of the above parameters for I+ 25. An abbreviated form of the final output results with each complete iteration. The final output has the total reaction cross-section resulting from s-wave alone, from s+ p-waves, from s + p+ d-waves, and from s+p+d+f-waves, if LMAX =3. The tensor polarization parameters are printed out for each angle, as are the experimental and calculated 73 2 ; . cross-sections. In addition, ax for the fit is given and the final DELTA(I) is presented in a column array. The coding for the program follows: ~73a- CHE(OIHE CONTROL PROGRAM “SJOIMENSTON 0D(25),SX(50),SA(59),S5L (50,25) ,DELTA(25) sSIGEX( 25), { 9 2u0 2cl L3u 131 WIG 91 él Lol lil WIG 92 &2 Lo2 112 LOOELTAC25},SI1GCA(59}) , ANGLE (5G) SLOPE (53,25) ,2159) RISD), 2° (19,50) ¢P1015_,50) 9P21105350) ,TEMP(50) sDELTAZ(25), ZBELTA(25)¢V(10),S1GCAS(59,25),DELTA3(25) ,DDELT3(25), 48650) 58159720150) ,F150) E159) ,AL150)},81150),C1150) -D11(50),E1(50), 5AZ(59),82159) ~C2(50) 202159) 2E2(50),43(50) .831(50} 5C3(50),03150), 603(59),U105),0U265)5U305) 40405) 50505) .SANG(50} eSANG2(50),SANG4G(50) TCANG(59} sCANGZ{ 52), TRHOI5C), 711150) 57T22(59) 9T21050)5729159) DIMENSTON ALPHA(17),WAVENO(2} JCOMMON Oy, I MAX sNMAX, SAR oP ePLyP2yLMAXs Yo KW WHAVENG > ANGLE LAsBeCeF eo Es Ul pU2¢U3,U42U5, SANGZ, SANGS SQEQUIVALENCE (NELTA.I), CIMAX,MAKI), (NMAX,MAXN),(SIGCA,SA), LUISIGEX,SX),ESLOPE,SLI ee (WOT2W) s(CHISOQ,CIO) NREAD INPUT TAPE 55200 sNMAX,IMAXsKWeWGTs Te (SIGEXIN) »ANGLEIN)» LN =1,NMAX) [5 =IMAX/2 16 =26 I? =25+IMAK/2 READ INPUT TAPE 5,291,LMAX,CDELFACI READ INPUT TAPE 5,201,MAXL,(DELTACI FORMATU314%,2F15.5/€2F10.5)) FORMATUI4/(F19.5)) DO 130 T=1,15 CELTACT +25 )=OELTAC1+253/57.295780 DELTAC(I) SDELTA(I)/37.295780 NO 131 N =LyNMAX ANGLE(N) =ANGLEIN)/57.295780 GLE REAL PARTS OF DELTAS L=0-2 AND EPSILON 1 (8 PARAMETERS} IP} 1 IP2 IMAX/2 IP3 1 GO TO 1 50 TO (21,31552).KKW GO TG 2 60 TO 4 GLE IMAGINARY PARTS UF DELTAS L=0-2 AND EPSILON 1 (8 PARAMETERS} [Pl =26 IP2 =25+IMAX/2 [P3 = 1 GO Ta l GO TO (21%31152)_9KKW GO TO 2 GO Tu 4 dot dy! i,I5) To6;17) WIGGLE REAL AND IMAG. PARTS OF DELTAS L=G-2 AND EPSILON 1 (16 PARAMETERS) 93 IP1=26-IMAX/2 IP2 =25+IMAX/2 IP3 =1 co Ta 1 83 Tl =IMAX/2 00 883 I =1,11 1P4 =25-IMAX/241 883 OC FPS) =O(T) GO TO (21,31952)5KKW 103 00 903 [=12,1T1 1P4 =25-1IMAX/2+1 903 OCL)=OCIP4) -73b-. GO TO 2 1143 IL=IMAX/2 DO 913 [=l,I1l ~TP4& =25~-IMAX/2¢] 00 913 N =1,NMAX 913 SLUN,IP4)=SLIN,1) GO TO 4 WIGGLE REAL AND IMAG. PARTS OF J=lylL=1,DELTA ({2PARAMETERS) 94 IPL = 3 [P2 =28 IP3 =25 GO TO l 84 GO TO. (21531,52),KKW 104 GO TO 2 114 GO To 4 . WIGGLF REAL AND IMAG. PARTS OF J=3,L=3,DELTA (2 PARAMETERS) ana 95 IPL = IMAX/2 IP2 =25+I1MAX/2 IP3 =25 GA TO 1 85 GO TO (21431,52),KKW 105 GO TO 2 115 GO TO 4 WIGGLE REAL AND IMAG. PARTS OF J=3yL=3,J=29L=3,J=2,L=1, DELTAS (6 PARAMETERS) 96 IPL =36 IP2 =41 IP3 = 1 GO TO l 86 D(36) =D(9) D(37) -D(10) D(38) =0(2) D(39) =D0(34) D(40) =0(35) D(41) =D27) GO TO (21,31,52),KKW 106 N(9) =D(36) N16) =Nt37) D(2) =0(38) D(34) =0(39) 0(35) =D(49) O27) =D0(41) GO-TO 2 116 00 126 N = 1,yNMAX SLIN,36) =SLON,9) SLIN,37) =SLOIN,19) SLOIN,38) =SLIN, 2) SLIN:39) =SLUN, 34) SE(N,40) =Si(N,435) 126 SLOINs41) =SLIN,27) GO TO 4 WIGGLE REAL AND IMAG. PARTS UF J=3,L=3,J=2,L=3,J=2,L=1, DELTAS AND EPSILON 2 (8 PARAMETERS) 97 IP1l =37 [P2 =44 IP3 = 1 GO TO 1 87 0(37) =D(9) 0(38) =D(10) 107 127 C WIGGLE REAL AND IMAG. D(39) =D0¢11) 040) =0(2) D(4l) =D(34) Dt42) =Dt35) Di43) =0(36) D(44) =0(27) ~73c- GO TO (21,431,521) 4KKW D(9) =D(37) D(10) =0(38) N(11) =0(39) Dt2) =0¢40} 00434) =O0t41) D(35) =D42) D(36) =0(43)} 0127) =0(44) GO TO 2 DO 127 N = 1_NMAX SLUN,37) =SLIN,9) SLON,38) =SLIN,10) SL(N,39) =SL(Ne11) SLUN,40) =SLUN,2} SLUN, 41) =SLIN, 34) SLON,42) =SLIN, 35) SLON,43) =SLCN; 36) SLON,44) =SLON227) GO TO 4 PARTS J=3,t=3,J=2sl=3,J022,t1lyJ=let=) C OELTAS AND EPSILON 2 (10 PARAMETERS) 98 88 108 118 IPl =37 IP2 =46 IP3 = 1 Go TO l 0(37) =D(9) 0(38) =0(10) DI39) =D11) D(40) =DI2) 0(41) =D{3) Dt(42}) =D(34) D(43) =Dt35) 0(44} =0(36) 0445) =0¢27) O(46) =D(28) GO TO (21,31,52)sKKW DI9) =0(37) O(1G) =0(38) DI{11) =0(39) D(2) =D(49) Dt3) =D(41) 0(34) =D142) 0135) =Dt43) D(36) =0144) Dt27} =Dt45) D(28) =0(46) ‘60 TO 2 00 128 N u L»>NMAX SLIN,37) =SLINs9} SLON, 38) =SLUN,10) SLIN, 39} =SLUN~ 11) SLUIN,40) =SLOEN, 2) aM -73d- SLINS41) =SLIN,3) SLON, 42) =SLOUN, 34) SLIN,43) =SL(N,35) SLON,44) =SL(N,36) SLIN,45) =SLINy27) 128 SLIN,46) =SL(N,28) 60 TO 4 ; WIGGLE REAL AND IMAG. PARTS OF ALL DELTAS THRU L=2 AND ALSO $=3,.=3, J=2,L=2,0ELTAS PLUS EPSILON 1 AND EPSILON 2 99 IPL =26-IMAX/2 IP2 =25 +IMAx/2 IpP3 =1 GO-TO lL 89 [1 =IMAX/2 DO 889 [ =l,tl IP4 =25-IMAX/2+] 889 DLIP4)=D(T1) GO TO (214531,52),KKW 109 DO 909 [=l,I1l IP4 =25-IMAX/2+41 969 NCL) =O(1P4) GO TO 2 119 T1l=IMAx/2 NO 919 T=1,11 IP4 =25-IMAX/2+I DA 919 N =1_yNMAX ILI SLIN,IP4S)=SLONG IT) 60 TO 4 : 1 CHISOI = 20, : BETA =SQRTF((2e8TI/(2.89316141) )#01.-(308T) 164042 .49312.141)) wWAVENOCL ID =( 20846 8BOTA#LE-L2"93 Le LSLIS (Le PTIE~“LILO(Z0t4e)?) WAVENO(2) =0.9 ETA =(2241.44E-13)/(1.973E-LL#BETA) DO 252 N=1l,sNMAX ANG =SINF(ANGLE(N)/2.) ANG 2=ANG#ANG TEM =ETA#LOGF(1./ANG2) Nl =N+4+50 ; ANGLE(NL) =9.9 Z(N)=COSF(TEM) Z(N4+50)=SINFCTEM) TEM=-O0.5*#ETA/ANG2 R(N)=TEM#Z(N) 252 R(N+50) = TEM#7Z(N450) JMAX =LMAX +1 JP2=JSMAX 42 DO 253 J =2,JP2 JP =Jj -1 ALPHACL) =O 0 253 ALPHALJ)=ALPHA( JP) 42.24 TANF(ETA/FLOATF(JP)) CO 254 N =lyNMAX SANG(N)=SINFCANGLE(N)) SANG2(N)=SANG(N) #®SANGEN) SANG4(N) =SANG2UN) #SANG2(N) ’ CANG(N) =COSF(ANGLE(N)) CANG2(N)=CANG(N) #CANG(N} SPZ=JMAX44 P(1lyN) =1.0 P(2,N)=CANGIN) -73e- DG 254 5$=3,J5P2 Jl =J-1 J2 =J+l1 J3 =J=2 AJ =J AJl =J1 254O0P CS NIFC2eHALLAL.D/AJILECANGIN) @PLILINI-LASL—“1Le)/ASLEP (IZ 9N) DG 255 N =LyNMAX Pl(1,N) =0.0 Pl(2,N) =1.0 JP2=JMAX43 DO 255 J =3y5JP2 Jl =J-1 J2 =J+1 AJ = AJL =J1 2550P LOS sNI=CASL#(ASLEL.I/ (20 8AIL 41.) HIP (SL PNI~POS29N) )/01.-CANG2(N)) 256 20 31 23 25 43 39 24 DO 256 N=lyNMAX P2 (1,N) =9.0 P2¢2,N) =0.0 P2(3,N) =3.C JP2=JIMAX42 DO 256 J =4,JP2 J1 =J-1 J2 =J+1 AJ =J OP2USeNISAIL#CASLELSI/(2.FAILEL. De CPL OSL eNI-PLOS2 NII CL.-CANG2(N}} 1+(2.*#CANG(IN) #PL(SyNDD/C2.-CANG2Z(N)) MAXJS =LMAX+3 DO 257 J=lyMAXS TEM=ALPHA(J)/2. Y(J)=COSF(TEM) Y(S+10)=SINF (TEM) KKwel KKK=1 RMS1=100.0 CALL XSECT GO TO(8, 8s 8585898, 8y 898, 8_9S8e98y8y100) pKKK KKK =KKK+1 CIQ=0.9 PO ZG NELyNMAK CIQ=CIQ+((SXEN)—SAUN) )/CWESXON)))#COSXEN)-SA(N))Z CWeSXON) )) GO TO (22,30),KKw GO TO(81,82583, 8%, 85, 86587788499) yKW KKW=1 : IF(CIQ-CHISQI}22922,23 TFCABSF(CIQ-CHISQI)-.05)22,22,25 O(33)=-D133) DEL TA3(33)=-DELTA3(33) ODELT3(33)= -DDELT3(33) DO 24 [=IP1l,IP2,1P3 DDELTACI)=.333"#D0ELT3(1) DELTAC I) =DELTA3UI)+0DEL TACT) ODELT3(1)=ODELTA(CI) TF(1-26)24,43,43 TF(D01))39,24,24 D(1)=0.0 CONTINUE QELTA3(33)=-DELTA3{(33) -73f- DDELT3(33)= -DDELT3(33) 0(33)=-D(33) KKW=2 GO TO €10151624103410%,105,106,107,108,109) KW C SLOPE 22 00 259 N=1l-eNMAX 259 TEMPIN) =SIGCAIN) 15 =IMAX/2 DO 260 [=lyI5 IF(ABSF(DELTA(T) )-.01/57.295780) 32533433 32 ELTAC(T)=.01/57.295780 GO TO 34 33 ELTACT)=1L.91#0ELTA(T) 34 TFC ABSF(DELTA(T +25) )-.61/574295780) 355 36% 36 35 ELTACI4#25)=.91/57.295786 GO TO 37 36 ELTACT4+25)=1. GLeDELTA(1+25) 37 DELTA2(1+25)=DELTAC(I +25) 260 DELTA2Z¢(1I)} =OELTACT) DO 262 ILM=1,15 DELTA( ILM+25)=ELTACILM+25) CALL XSECT DG 264 NIL=lLy,yNMAX SIGCAS(NILs ILM+25)=SIGCA(NIL) 2640SLOPE(NIL, ILM425)=(SIGCAS (NIL, ILM+25)-TEMP(NIL))/(DELTACILM+25)-DE ILTAZCILM+25)) DEL TACILM425)=DELTAZ(ULM425} DELTACILM) =ELTACILM) CALL XSECT DO 261 NIL =1,NMAX SIGCAS(NIL»sILM) =SIGCACNIL) 2OLISLOPEUNIL, ILM) =CSIGCAS(NILS ILM) -TEMPINILIJ/(DEL TAC ILM) -DELTA2ZCL 1LM)) 262 DELTACILM) =DELTA2ZCILM) 16 =26 17 =25+IMAX/2 0G 263 N=1,NMAX 263 SIGCA(N) =TEMPIN) GO TO CLLLs1L129113511491159116,117-118e119) Kw 4 DO ll I=IP1,I1P2Z,1P3 TEM=0.0 DO 1G N=l,NMAXx 10 TEM=TEM+SLIN,IT)#SLINST) DOELTA(1I)=9.0 DO 11 N=1,NMAX Ll DDELTACT)=DDELTACI)4(SKON)-SA(N}) #SLOINGTI/STEM DO 15 I = IPl,IP2,1P3 15 SUM = SUM + DDELTA(I) ## 2/FLOATFC(I(IP2 - IP1LI/1P3) + 1) RMS = SQRTF (SUM) IF(RMS-~RMS1)50250,51 50 RMS1=RMS GO TO 54 51 KKWL=KKW KKW=3 GO TO (815824835845 855 865 87598189) KW 52 0(33)=-Of 33) DELTA3(33)=-DELTA3(33) ODELT3(33)= -DOELT3(33) 49 44 53 54 21 204 40 41 4&2 6 75 T6 45 46 77 65 47 48 &6 73ag-. DU 53 JEIPLsIP2,51P3 DDELTA(1I)=.333#00ELT3(1) DELTAC(T)=DELTA3Z{I)+0D0ELTACI) ODELT3( 1} =ODELTAC IL) 1F(D(1)144,53,53 O(1)=9.0 CONTINUE DELTA3(33)=~NELTA3(33) DDELT3(33)= -NOELT3I(33) 0133)=-D( 33) KKW=KKWL GU TUC1LO1,102,103,104,195,106,107,108,159) , KW GO TO (81,82,83,84%s) 85, 86, 87488589) KW WRITE OUTPUT TAPE 6,203,CHISQ,(DELTA(I) -1=1,59) WRITE QUTPUT TAPE 6,204,RMS,(ODELTA(I),1=1,5C) FORMAT(F10.5/(F10.5)) TFORMS~.1/57.295780)100,100.40 TFCRMS-36/5724295780) 42,42, 41 IF ORMS=20./57.295780) 65,65,60 KKW=2 GO TO 65 O(337=-DU43) DEL TA3(33)=-DELTA3( 33} DDELTA( 33) =-ODELTA(33) DNELT3(33)= -ODELT3(33) CO 77 L=IP1L,1P2,1P3 IF (ABSF(DDELTA(I))-106/57.295780) 77, 77,75 IF(DDELTACI)) 709764 76 DELTA3(1)-O(1) NDELT3(1)=-3./57.295780 OCL)=D01)-3./57.295780 TF(1-26)77545345 DELTAS(T)=DtU1) DDELT3(1)=3./57.295780 OUI )=D(1)+3./57.295780 IF(1-26)77,45,45 IF(D(1))46,77,77 O(1)=90.9 CONTINUE DDELTA(33)=-D0ELTA(33) DEL TA3(33)=-DELTA3(33) DDELT3(33)= -DOELT3(33) 0(33)=-0(33) GO TU 3 0(33)=-D(33) DDELTA(33)=-DDELTA( 33) DO 66 T=I1P1,1P2,1P3 DELTA3(1)=01) ODELT3(1)=ODELTA(T) D¢1)=OC1)+DDELTACT) IF(1-26)66,47,47 IF(OU1)) 48,6666 N(1)=0.0 CONTINUE DELITAS(33)=-DELTA3(33) NDELT3(33)=-DDELT3(33) 0(33)=-D(33) CHISQISCIQ oO et et et tet ee = pt et I -73h- — GO TO (101,102,103,154,105,1064107,108,109) y KW 169 00 132 T=1,15 DELTA(I+25)=57.295780#DELTA(I+25)} 132 DELTACI) =57.295780#0ELTAII) DO 133 N =1,NMAX 133 ANGLO(UN) =ANGLE(N) 4#57.295780 POLARIZATION TENSORS 00 301 N=1,NMAX ALIN) =A{(N) BLIN)=BIN) CLIN} =(.707,0.G) *#SANG(N) #CON) DLN) =0.707,9.0) @SANG(N) #F ON) ELIN) =(.707,0.0) #SANG2Z(N) #E(N) A2(N)=ABSF(ALI(N)) B2(N)=ABSF(B1(N)) C2(N)=ABSFI(CLIN)) CAIN) =ABSFICDLIN}) C2UN)=ABSF(EL(N)) TRHO(IN) =(46674000)8(A2(N) #A2(N) 462550200) FB2(N) #B2IN) +5°2EN) #C2UN) + LOZ(N) #D2ZEN)+E2(0N) #E2(N)) DLIN+59)=-D1l(N+5G6)} EL(N+59)=-E1(N4+59) B3(N)=BLIN) #O1ON) C3INISCLONI®ELON) EZS(NI=ALCNJ #210N) CLl(INt59)=-C1(N+50) AZ(NJ=AL(N)J#C1ON) TLLOND=€1.414/ 01.7328 TRHOCND D(C BIINt5S9O) +C3INTSD) FAS (NT5O) ) T22(N)=0L./(1.732*TRHO(N) D8 (22 8ES(N)~OL(N) #DL(N)-D1L(Nt50)2D1L(N+50 1)) ‘ T2LIN)=-01.6414/(1.732*TRHOCN) )) #(B3(N)4C3(N)-A3(N)) SGL TZOUNJ=UCL 414%, 0.0) 71003. 9900) #TRHOIN) )) #(AZ(N) MAZ(N)4+02EN) #D2(N) + LEZ(NJ#EZ(NI—C2epIeOD AC2IND RZD 2(N)-BZ2(N) #B2(N) ) REACTION XSECT IPLG=LMAX+41 SIGRE=(0.0,0.0) CO 362 T=1,1P16 UVA=ABSF(UL(T)) UB=ABSF(U2(1)) UC=ABSF(U3(1)) UD=ABSF(U4IT)) UE=ABSF(U5(1I)) X=I-1 OSTGRE=SIGRE+(0 (321415009) /(WAVENO#WAVEND) 2 800 ( 209000) Xt Lee De DIU Loe GeO XH (Sey CoD) VS (309 GeO) MUCHUC (1205000) X+(1050260))/130,0.0)% ZUBEUB=1 00205769) BX CL ee FeO) ) S036 2900) ) MUA BUA—(( (209000) "X4+130 50.0) 3) /1354000)) UDRUD~-( 60205060) #X— (165920) 971 302000) ) #UERUE) 302 WRITE QUTPUT TAPE 6,363,SIG2E,(ANGLE(N) »TLICN) ,T22(N) ,TZLIN) ,T2ZO(N: 1),N=1,NMAX) : WRITE OUTPUT TAPE Ge2029Ts{ANGLOIN) pSIGEXIN) pSIGCAIN)D » N=lLyNMAX) WRITE OUTPUT TAPE 6,203,CHISQ, (DELTACI),1=1,50) 202 FORMAT (1LH1L25X20HPHASE SHIFT ANALYSIS///15X29HENERGY OF INCIDENT P LARTICLE =F10.5///2X20HCENTER OF MASS ANGLE2X26HEXPERIMENTAL CROSS 2SECTION2X24HCALCULATED CROSS SECTION///(5XF10055 LOXF10.5,18XF10.5) 3//7/) ZU30FORMAT(SXTHCHISQ =FLO.5///25X18HTOTAL PHASE SHIFTS///(30XF10.5) } JOIOFORMATCULHITASXIGHTENSOR POLARIZATION/S//15X2Q9HTOTAL REACTION CROSS §S LECTION=F10.5///2X20HCENTER OF MASS ANGLE7X3HT119X3HT229X3HT21 9X BHT 22IS TI USXF10 654 OXFL0 6 59 2XFL00 5 yg 2XFL0O05 y2XF1065)) 60 TO 9 END ee ee eee ee) — a ee re et — ee PR Pe -73i- SUBROUTINE XSECT SOTMENSTON AA(5) 4AB(5) ¥AC(5)98A(5),BBR(5) p6C15) CAS) »CB(5),CC(5), 10A4(5),0B(5),DC(5) .EA(5)yEB(5) 5EC (5) 4U1(5) .U2(5) »U3(5) ¥U4(5) pU505) 22(25)44(590)48(50),0(50),F (59) ,E(50) ,DELTA(25),SIGCA(50).ANGLE(SO} BRUSO}eP C190, 59),P1(10,50) »P2110,51),¥(19)»SANGZ(50) ySANGG(5Q) COMMUN DELTA, IMAX,NMAX,SIGCAsRyPyP1yP2,LMAXsY yKW ep WAVEND, ANGLE, LAgbe Cg Fy EF eU1L,U25U3,U4,U5, SANGZ,SANGS DO 207 N =1eVMAX Nl =N4¢50 JP2 =LMAX 41 DO 2C7 JS =LleJdP2 P&JdeNL) =Je) PI(JsNL) =0.0 P2(JeN1) =0,90 SU TOUlslelsyly4s5969646),KW O00 1G I =i,7 IP =f #25 Z(1I) =-2.*DEL TAC IP) 00 11 I =26.32 IP =I -25 ZtI1) =2.*DELTALIP) Z(8) =DELTA(#) 2(33) =DELTA(33) IP9 =3 U3(2) =EXPF(Z(2)) yvet2) =EXPF(Z2(3)) UL(2) =EXPF(Z(4)) U42) =(20995.9) U5(2) =19.9,0.9) u3(3) =EXPF(Z(5)) U2(3} =EXPF(Z(6)) CUS=COSF(2Z(8)) SYN=SINF(Z(8)) XPL=EXPF(Z(1)) XPT=EXPF(Z(7)) UL(3)=(SYN#SYN) ® XPL+ CUS#CUS# XP7 U4(3) =(9.3,0.0) USCS RHF edeVeCIFISINE (Coe yUeDD ALIS) PIL XPI- XP) U3(1)=CUS#CUS# XPL+SYNESYN® XPT U2(1) =(0.590.0) UL(L) =€9.9,0.9) U4(1)=U5(3) DO Lud N=1,1°9 x =N-1 AACN) =OOX+E2 er Oe DDRUBIND 102090. DEX ( Le pe) ) FUZINI+(X—-C 10900) ) PUL LIN HC2er De HLM 2ee Oe AX +1 6206))) ABIN) =SQRTFUUK+E Ls pGo) BIKE 205 G0 dd) PUSIN) ALIN) =SQRTFCKPCATCLe pte DDD FUSIN) BACND=OX#( Ler Oe) LEUZIND EXHULINI 0020900) AXtH (lee de)) BB(N) =SQRTFUCX# Le te) DeIXH (20960) DD FUGIN) BC(N) =SQRTF(X#(X-€ 1.90.) ))#U5S(IN) ~ CAEN) =OX#OXF (26 pOe DP EUZINI “1 (209 Ge)HXt (10500) HUZIN)-(X##2—(1o 70. LIV#ULOND I /OXe(X+C 1.20.) )) CBIN) =SQRTFCCX+O Le pCod IS OXHC 20790 dF) RUGIN) CCIN) =SQORTF (XS (X-01.2,2.2))) 8U5 ON) DA(N) =U3(N) -ULEN} DB(N) =SQRTFUCX#C Le gC ed IAC XH 20500) )) HU4UN) DO(N) =SQRTFUX/(X-€1. 950) ) }RUS(N) 73 j- I CACND=CX#UBIND=C 0 2ep Cod OXH( Le p00) PHUZIND H1XH( Lee Oe) DMULENDISIX® L(X#(1.500))) I EBIN) =U4(N)/SORTF((X+( 1.200). #1X4(2090.))) I Lod EC(N) -=U5(N)/SQRTFUX#(X-(1.900))) GO TO 50 C CASE 5 4 DO 2U I =1,7 IP =1 #25 25 ZL) =+2.*DELTACIP) DO 21 I =26,32 IP =I -25 21 Z(1) =2.8OELTALIP) Z(8) =OELTA(8) Z2(9) =-2.#DELTA(34) 2133) =DELTA(33) 2034) =2.*#DELTA(9) te eee U3(4) =(9.9,9.9) U2(4) =EXPF(Z(9)) U164). =(0.0,%.0) U4lG) =00.99 7.0) U5(4) =(9.6,0.0) Ip9 =4 GO TU 2 C CASE 6 5 DU 3u I =1,7 Ie =I +25 39 ZT) =-2.#DELTALIP) O00 31 1=26,32 [Pp =I ~25 31 ZOE) =2.*DELTACIP) Z2(8) =OELTAt8) Z(9) =-2."DELTA(34) Z(10) =-2.*#DELTA(35) Z(33) =DELTA(33) 2(34) =2.*8DELTA(9) Z2(35) =2.*#DELTA(19) I u3(4) =(0.9,9.0) i U2¢4) =EXPF(7Z(9)) I U1(4) =EXPF(2Z(10)) I U4(4) =(9.0,0.0) I U5(04) =(0.0,%.0) IP9 =4 GO TO 2 C CASES 7-9 5 00 40 [T =1,7 IP =1 +25 40 Z(1) =-2.#DELTA(IP) DO 41 1=26,32 TP =1-25 41 Z(T) s2.*#DELTACIP) Z(3) =DELTA(8) Z(9) =-2.*DELTA( 34) Z01C) =-2.*DELTA(35) Z{11) =DELTA(11) 2133) =DELTA(33) Z2(34) =2.*#DELTA(9) 7(35) =2.*#DELTA(1G6) 2(36) =DELTA( 36) I SYN=SINF(Z(¢11)) eel email nll nla on Mn Ee on oe Ta Mee Hl ool om Bian (onal om fice Eon El on aoe 51 52 530 690 61 -73 k- CUS=COSF(2Z(11)) XP2=EXPF(Z(2)) XPLO=EXPF(Z(10)) U362)=CUS#CUS® XP2+SYNHSYN#® XPLO U2Z(2) =EXPF(Z(3))} U1L(2) =EXPF(Z(4))} U4(2)=0.50,0,0)#SINF((2e,0.0) #2011) ) #4 XPLO-— xP2) U5(2) =(9.0,9.0) U3(4) =(0.0,06.0) U204) =EXPF(2(9)) U1(4)=CUS#CUS#® XPLOFSYNSSYN® XP2 U4(4) =10.0,%.0) U5(4)=U4(2) TPQ 24 GD TO 3 3 SMAX =LMAX +1 NO 60 N =lyNMAX ACN) =REN) BIN) =RON} CIN) =(9.0,0.0) F(N) =(0.0,0.0) EIN) =(0.050.20) O00 69 J =1,JdMAX K =J +2 IF(J-2)52,52,51 M =J-2 GO Tu. 53 M =1 ACN =AUCND FTV OSS C0 A Oe ee DEC VCS OPEN) KAAC II TY CK DP LK N) MABC UIT VEIM) L#P(M,N)#AC(J)) SBIND=BENDF¥0SI/(90 20,20) 81 VY (SDP (J eN) #BACJI-YOEK) @P(KeN) #BB (9) L-Y(M)#P(MsN) *#BC(J)) ICINISCIND FV CSIC Oe Zed CVI SDHPL ISIN ACACJI-Y CK) OPLAK NI SCB(S) L+¥(M)#P1(M,N)e#CCOJ)) DEON SFOND +Y¥OS) 700009 20 CVC SD RPL OS N) HDAC SDV OK) @PL (KON) FOB) 1-YOM)#PLIM,N) #O0C09)) SEINJ=EIND +Y(05)700.952.828) #0 VCS) P20 SeN) MEACJIFY COKE eP2(K ND HEB) L+Y(M)#P20M,NI #EC(J)) DO 61 N =1,NMAX AX=A8BSFCA(N) } BX=ABSF(B(N)} CX=ABSFICIN)) DOX=ABSFCFIN)) EX=ABSF(E(N)) SIGCAIN) =0(2. 500) MAX PAX+BXHBX+SANGZ(N) #(CX#CX4#DX8DX) +SANG4UN) #EX8 LEX) /(03490.) *WAVENO#WAVEND) RETURN END 74 REFERENCES F. Ajzenberg-Selove and T. Lauritsen, 1959, Nuc. Phys. ii, 1. K. W. Allen, E. Almqvist and C. B. Bigham, 1960, Proc. Phys. Soc. (London) 75, 913. J. G. Allred, D. K. Froman, A. M. Hudson, and L. Rosen, 1951 Phye. Rov. 82, 786. V. V. Balashov, 1959, Soviet Physics-JETP 9, 798. A. M. Baldin, V. I. Goldanskii, andI. L. Rozental , 1961, Kinematics of Nuclear Reactions (Oxford University Press, London), J. M. Blatt and L. C. Biedenharn, 1952, Revs. Modern Phys. 24, 258. J. M. Blatt and V. F. Weisskopf, 1952a, Theoretical Nuclear Physics (John Wiley and Sons, New York). G. Breit, H. M. Thaxton, and L. Eisenbud, 1939, Phys. Rev. 55, 1018. E. J. Burge, H. B. Burrows, and W. M. Gibson, 1952, Proc. Roy. Soc, (London) A210, 534. E. U. Condon and G. H. Shortley, 1935, The Theory of Atomic Spectra (Cambridge University Pross, Cambridge, England). W.C. Dickinson and D. C. Dodder, 1953, Rev. Sci. Inst. 24, 428. R. D. Evans, 1955, The Atomic Nucleus (McGraw -Hill Book Company, Inc., New York). A. Galonsky, R. A. Douglas, W. Haeberli, M. T. McEllistrem, and H. T. Richards, 1955, Phys. Rev. 98, 586. A. Galonsky and M. T, McEllistrem, 1955a, Phys. Rev. 98, 590. 75 J. L. Gammel, B. J. Hill, andR. M. Thaler, 1960, Phys. Rev., ily, 267. | W. Haeberli, 1963, (private communication). R. L. Henkel, J. E. Perry, Jr., andR. K. Smith, 1955, Phys. Rev. 99; 1050. R. D. Huntoon, A. Ellett, D. S. Bayley, and J. A. Van Allen, 1940, Phys. Rev. 58, 97. D. R. Inglis, 1952, Phys. Rev. 87, 915. D. R. Inglis, 1953, Revs. Modern Phys. 25, 390. W. Lakin, 1955, Phys. Rev. 98, 139. A. M. Lane andR. G. Thomas, 1958, Revs. Modern Phys. 30, 257. T. Lauritsen, T. Huus, andS. G. Nilsson, 1953, Phys. Rev. 92, 1501. T. Lauritsen and F. Ajzenberg-Selove, 1962, Enerey Levels of Light Nuclei (National Academy of Sciences--National Research Council, Washington, D. C.). H. W. Lefevre, R. RK. Borchers, and C. H. Poppe, 1962, Phys. Rev. 128, 1328. J. B. Marion and A. S. Ginzbarg, 1949, Tables for the Transformation of Angular Distribution Data from the Laboratory System to the © Center of Mass System (Shell Development Company, Houston, Texas). E. M. Pell, 1960, Jour. App. Phys. 31, 291. R. J. N. Phillips, 1960, Proc. Phys. Soc. (London) 75, 317. 76 L. G. Puondrum, 1959, Phys. Rev. betters a; 346. L. Puzikov, R. Ryndin, and J. Smorodinsky, 1957, Nuc. Phys. 3, 436. G. Racah, 1942, Phys. Rev. 62, 438. J. L. Russell, Jr., G. C. Phillips, and C. W. Reich, 1956, Phys. Rev. 104, 135. B. V. Rybakov, V. A. Sidorov, and N. A. Viasov, 1961, Nuc. Phys. 23, 491. R. G. Sachs, 1953, Nuclear Theory (Addison-Wesley Publishing Company, Inc.) Chap. 10, Appendix 3. F. Seiler, S. E. Darden, L. C. McIntyre, and W. G. Weitkamp, 1962, Bull. Amer. Phys. Soc. 7, 531. E. A. Silverstein, 1959, Nuclear Inst. and Meth. 4, 53. H. P. Stapp, 1957, Phys. Rev. 107, 607. L. Stewart, J. E. Brolley, Jr., and L. Rosen, 1962, Phys. Rev. 128 , 707. T. A. Tombrello, 1963, (private communication). P, H. Wackman and N. Austern, 1962, Nuc. Phys. 30, 529. W. Whaling, 1958, Handbuch der Physik 34, 193. E. P. Wigner, 1932, Gbttingen Nachr. 32, 546. K. Wildermuth, Y. C. Tang, F. C. Khanna, andt.. TD. Pearlstein, 1961, Proc. Rutherford Jubilee Int. Conf. (Manchester) p. 259. H. R. Worthington, J. M. McGruer, andD. E. Findley, 1953, Phys. Rev. 90, 899. 77 Table I. Energy corrections for deuterium gas and 2500 A nickel foil, with alpha particles incident. See text, p. 23. Nominal Energy Corrected Energy (Mev) 20 cm. -oil 25 cm. ~oil (Mev) (Mev) 6.000 5.875 5.865 7.000 6. 887 6.878 8.000 7.895 7.887 9. 000 8.904 - 8.897 10.000 9.909 9.903 11.000 10.916 10.910 12.000 11.920 11.915 78 Table II. Energy corrections for helium gas and 2500 A nickel foil, with deuterons as the incident particles. See text, p. 23. Nominal Energy Corrected Energy 20 cm. -oil 25 cm. -oil {MeV) (MeV) (MeV) 3.000 2.958 5.000 4.973 4.971 6.000 5.976 5.973 7.000 6.979 6.977 8.000 7.981 7.979 9.000 8.982 8.981 10.000 9.984 9.982 11.000 10.984 10.983 12.000 11.986 11.984 79 Table Ill. Energy corrections for helium gas and 5000 A nickel foil, with deuterons as the incident particles. See text, p. 23. Nominal Energy Corrected Energy 20 cm. oil 25 cm. «oil (Mev) {MeV} (MeV}. 5.000 4.956 4.954 6.000 5.962 5.960 7.000 6.966 6.964 8. 0U0U0 7.969 7.967 9.000 8.971 8.970 10.000 9.974 9.972 11.000 10.975 ~ 10.974 12.000 11.977 11.975 Table IV. Systematic Errors ae Error in Geometric Factor Gs 2bA/R oh Parameter 2b awe > b. Error in Charge Collection N = Vt/R Parameter V t R N c. Error in Particle Density na (S.G.)p/T Parameter 5.G. Pp T n 80 Error (percent) 4 25 - 005 21 . 648 RMS Error (percent) +02 202 ol - 1039 RMS Error~percent) 25 ow) od 0574 d. Error in Approximating Cross-Section Equation Parameter pn?) 3h" RMS Error (percent) 0948 Probable angular independent systematic error = 1.0 See text, pp. 25ff. 81 Table V. Error due to sin4) | (do, »/4 cm! Sab (degrees) RMS Error (percent) 12.0 1.5 15.0 1.0 20.0 05 25.0 5 30.0 «5 | 40.0 25 50.0 5 55.0 25 60.0 05 65.0 5 70.0 5 80.0 25 90.0 25 See text, pp. 27ff. Table VI. 82 Error due to tan Sab g lab (degr ees) 12.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 See text, pp. 27ff. RMS Error (percent) 1.0 1.0 1.0 83 + Table VII. Single level parameters for the 2 level in Lic. The units of yo tabulated here differ from those described in the 2, text, p. 61 , in that the units of y in the text are MeVecm. Level 2* Interaction Radius 4.8 fermis Reduced Width 0.981 MeV Excitation Energy x 24.57 MeV Ratio to Wigner Limit 0, = yk > ) = 0.481 aa. 84 + Table VIII. Single level parameters for the 1 level in Lio, 2 The units of y tabulated here differ from those described in the text, pe 61° , in that the units of ye in the text are MeV-cm. Level \* Interaction Radius 4.8 fermis Reduced Width 1.005 MeV Excitation Energy KE = 5.62 MeV * 2 . are . 2 2 3h . Ratio to Wigner Limit 8, = Yo /¢ 5 ) = 0.492 aa 85 Table IX. Single level parameters for the {1 ) level in Lic. The units of y* tabulated here differ from those described in the text, pe 61 , in that the units of yin the text are MeV-cm. Level (1°) Interaction Radius 3.5 fermis Reduced Width 1.46 MeV Excitation Energy 7.19 MeV Ratio to Wigner Limit gs vl ( sn } = 0.380 aa Tensor Polarization Parameters at E Center of Mass Angle (degrees) 50 .00000 60.00000 70.00000 80.00000 90.00000 160.0000) 110.00000 126.00000 130.00000 See Appendix E. 86 Table X i(T11) “8.21539 -0.21781 -0.16706 -C.08516 -0.90000 0.07094 0.12328 0.15830 0.17668 d (T22) 217050 0.22571 0.25059 0.23897 0.20078 6.15332 C.10957 0.07475 0.04893 = 2.935 MeV (T21) —-0212616 -0.17298 -0.16429 -0.09963 -0.00000 0.10728 0.20168 0.27117 0.30878 (720). 0.17335 0.23764 0.24186 0.20701 0.16391 0213979 C.14972 9.19698 0.27625 Tensor Polarization Parameters at E Center of Mass Angle (degrees) 50.00000 60.00000 70 .00000 80.00000 90.00000 100.00000 110.00000 120.00000 130.00000 See Appendix E. 87 Table XI i (T11) -0.30038 -0.29115 -0.21385 -0.08587 G.04877 0.14725 0.19710 0.20910 0.19736 {T22) C.05590 C.04639 C.02105 -0.03115 -G.210771 -Ge18261 ~0.22575 -0.22595 -0.19162 d = 3.441 MeV (T21) -0.19473 -0.25485 -0.24731 -0.15457 0.00820 0.18910 0.33440 0.41733 0.43648 (T20) G.04515 0.06909 0.04863 -0.01056 -0.09101 ~0.14797 -0-13686 -0.04682 0.09989 88 See Appendix E. Table XIi Tensor Polarization Parameters at Ey = 3.946 MeV Center of Mass Angle (degrees) i (T11) (T22) (T21) «T20) 5§0.00000 ~0.46372 -0.14966 ~0.35867 -0.99362 60.00000 -0.46150 ~0.19379 ~0.4G6459 -0.17396 706.~00000 ~-0.36301 -0.19658 ~0.37163 -0.19346 80.00000 ~G.16429 -0.18326 -6.23215 -0.16124 90200000 0.05518 -0-223093 0.06390 -0.19194 100. 00000 0.18752 -0.34575 C.2569060 -0.30982 110.9000G 0.221918 ~0.424T1 0443492 —-Q.36225 -120.00000 0.220129 -0.408B9 Ge50740 ~0.226128 130.60600 | G.16986 -0.232561 0.49509 -0.05796 89 See Appendix E. Table XIII Tensor Polarization Parameters at Ey = 4,450 MeV Center of Mass Angle (degrees) i (T1121) (T22) {T21) (T20) 50.00000 -0.55956 -0434878 -0.47930 -0.18838 60.00000 ~0.57039 -0. 38696 ~0.-51028 -0. 36952 70.00000 ~0.43249 -~0. 20628 ~0.40434 -0.29023 80.00000 -0.19900 O- 11591 -0.19712 0.03626 90.00000 -0.03876 0.25528 ~0.00281 0.20536 100.00000 0.01260 0.10187 0.16581 0.04005 110.00000 0.02265 ~O.17146 0.32647 -0.22840 120.0000U 0.03653 —-02e33745 0243335 -0. 30605 130.00000 0.05270 -0. 33500 0644916 ~0. 15860 90 Table XIV Tensor Polarization Parameters at Ey = 4.731 MeV Center of Mass See Appendix E. Angle (degrees) i (TIL) (T22) (T21) {T20) 17297000 -0.14381 -0.01744 -0.04886 0.07407 22444009 -0.24087 -0.04998 ~0.11815 0.17655 29285000 -0.36355 ~0.12856 ~0.23415 0.21110 37220000 -0.45742 ~9.22611 -0.33416 0.11794 44..480C0 -0.53952 -0.33307 -0.41986 ~0.04554 §1.67000 -0.599C05 -0.42039 ~-0.48034 -0.23385 58.75000 -0.60164 —-0242357 —02e48781 -0236690 65. 70000 -0.51106 ~9.26829 ~0.41522 -~0.33179 72252000 -0.34561 0.02788 -0.28029 -0.09869 85.66000 ~G.08905 0.48752 -0.05260 0.37485 91.95000 -0.05243 0.50787 0.01017 0.40006 98202000 -0.05071 0.41304 0206491 0229412 4103.88000 -0.05991 0.24665 9.12719 0.11386 110.00000 -0.06419 0.04085 0.20455 ~0.08926 120200000 -0.04148 —9422434 0.32222 —0227404% 130200000 0.00016 -0.29733 0.36880 -~0.19572 140200000 0.02960 ~0.23688 0.34876 0.01390 150.000090 0.03862 —-0.14263 0.26719 0.21817 156.900000 0.03635 -0.09233 0.21606 0.31465 91 See Appendix E. Table XV Tensor Polarization Parameters at Ey = 4.955 MeV Center of Mass Angle (degrees) i (T11) (T22) (T21) {T20) 17.97000 ~G.15703 -6.02388 ~-0.05794 6.14590 22244000 —-0.25265 -0.05881 -0.11962 0.22921 29.85000 0.37442 ~0.13899 -0.21776 0.23193 37220000 —0.47219 -0.23839 -0.30271 0.12559 44.46000 -0.56014 —-0. 34730 -0.37571 -9.04124 51.67000 -0.62291 -0.43080 -0.42391 -0.22666 58. 75000 ~0.61728 -0-40954 —-0441754 -Q.34301 65.70000 ~0.50229 -0.19894 -0.33286 -0.26704 72252000 ~0.31469 0.15420 -0.20188 0.00957 85.66000 -~0.07796 0.63088 -0.03527 0.48255 110.00006 -0.12272 0.113603 0.10841 ~0.00559 120.00000 -0.10398 -0.132938 0.21667 ~“0.24694 1306.00000 -0.04741 ~6.25566 0.28041 -0.22436 140.00000 0.00054 -6.222201 0.27329 -0.04236 156-09000 O.-92272 “0.13760 0.22154 0O-15093 156.0000 0.02567 ~9.08993 0.18071 0.24399 92 Table XVI Tensor Polarization Parameters at E, = 5.473 MeV Center of Mass See Appendix E. Angle (degrees) i (T11) {T22) (T21) (T20) 17.97000 -G.17616 -0.03055 -0.05391 0.19661 22244000 -0.26990 ~0.06473 -0.09147 0.23291 29.85000 -1.39031 -C.13888 ~0.14543 0.19265 37~.20006 -0.49130 -C.22973 ~0.19005 0.08717 44.48000 -0.57917 -0.32589 -0.22521 -0.05248 5167000 -0.62867 ~0.38378 -0.23963 -0-18946 58275000 ~U.58113 -0.30789 -0.21278 —0-23489 65. 70600 0.40329 ~0.02283 -0.14083 -0.0856G 7252060 -0-18526 0.35995 -0.06637 0.20430 79.180C6 -0.05075 0.64040 -0.03146 0.44861 8566000 -1.01762 G.76101 -0.03251 0.55836 91.95000 -0.04717 0.75913 ~0.04859 0.54616 9802000 -0.10397 0.67380 -0.06328 0244459 103.88000 -0.16422 0.52941 -0.06630 0.28281 110.00000 -0.21306 233187 -0.05022 0.07716 120.00000 “9.22122 000216 0.01672 ~0.20:796 130.G9000 -0.14736 -0.17820 0.08845 -0.27129 140.00000 -0.06399 -0-18630 0.11825 -0.15382 150.00000 0.91349 -0.12233 0.10872 0.00138 _ 156.00000 C.00111 -0.08098 0.09224 0.07951 93 Table XVII Tensor Polarization Parameters at Ea = 5.961 MeV Center of Mass See Appendix E. Angle (degrees) i (T11) {T22) {T20) (T20) 22244000 —0.27465 ~0205693 -~0.04938 0.14895 29285000 -0.38263 ~O0.1151C -0.06576 0.09252 37220000 0246943 “—O.18548 ~0.07581 0.00893 44248000 -0.253725 ~6.25742 -~0.07975 -0.08524 51.67000 ~0.54957 -0.28823 “Ol onans -0.15935 58.75000 -0.43812 -0.18637 -0.05252 -0.13836 65.70600 -0.19755 C.09853 -0.02913 0.03832 72252000 G.02250 0642313 -0.02546 0427472 79.18000 0.10795 0.62354 ~0.04409 0.42704 91.95000° 7.90891 0.67649 -0-10151 0.43178 100.00000 —-0.12241 G.57189 ~0.13272 029663 11¢.00000 -0.27810 0.33298 -0.14543 0.02698 120.00000 233636 0.04495 -0.10938 -0.23728 13¢.acc000 -C@.261432 ~C.13261 ~0.04358 ~0.32232 140.00000 “9.14524 ~ON5205 0.00477 -0.23818 150.00000 “1.06367 -9.10108 C.02303 -0.11422 156.00000 -7,03499 -0.06687 0.02410 -0.05170 94 Table XVIII Tanszor Polarization Parameters at E a = 6.476 MeV Centar of Masa See Appendix E. Anzle (degrees) i<Tld) (T22) (T21) (T 205 27.97900 "0.18068 —“Ge02227 -~0.01302 0.05515 22544000 =0.24960 ~G.04087 ~0.09818 6.0323¢6 29485000 "0.33008 ~0.07852 -0.009010 -0.61799 37.20500 0.39292 ~0.12441 Q@.01097 -9.06G941 44.48000 -0.43466 ~0.17138 0.02369 -9.11712 °$1467000 -0.40892 -¢.18537 0.03305 -9.13631 §8.75000 “0.23670 ~0.08908 0.02333 -0.07114 65.76000 — G.04822 C.214187 -0.01901 0.09101 72252000 0.23596 0.35870 ~0.07227 | 0.23353 79218000 0.25838 G.46220 -0.11182 0.28329 85.66000 0.19062 0.48338 -0.138633 0226492 91295009 9.08702 0.46174 70.15847 0.20716 98.02060 -0.03165 6.41362 -0.17540 0.12146 ~160.00006 -0.07317 0.39235 -Q0.18038 6.08720 103.886900 ~0.15708 0434205 ~0.18680 0.01112 116.00000 ~0.28800 0.23853 -0.219546 -0.12899 120.G0000 -0.42983 0.03146 -0.17254 -0 434820 130 .00000 ~6.38933 -~0.10782 -0.10857 ~0.40725 140.00000 ~0.24944 -0.11931 -0.04994 0.31640 156.00000 -0.13427 ~0.07629 -0.01846 -0.19170 156.00009 ~0.G3833 -0.049953 —-90-00917 ~00e13675 95 Table XIX Tensor Polarization Parameters at EB, 2 6.965 MeV d Genter of Mass See Appendix E. Angle (degrees) i <(T11) {T22) (T21) {20} 17.97000 0.15513 -0.01439 0.01118 °-0.03822 22444000 0.20254 -0.02473 0.02244 +0.06605 29.85000 0.25220 -0.04483 0.04245 -0.09829 37.20000 0.28652 -0.,G5912 0.06289 ~-0.11877 44. 48000 -0.30017 0.09326 0.08028 —0.12556 51.67000 ~0.24038 -0.09421 0.07797 -0.09769 58.75000 -0.@271) —9.01895 0.01970 —-0.00439 65.70000 0.25400 6.13003 0.08869 0.10957 72.52000 0.37399 0.23680 0.16566 0.14540 7918000 0.34511 0.28974 -0.19350 0.11390 85.66000 0.26249 G.26541 0.20028 0.05634 91.95060 0.16387 0+24558 0.20248 0.01348 98.02000 0.05519. 6.21851 0.20534 -0.09561 400200000 0.01612 0.26450 —0.220857 ~0.1263%8 163.88000 -0.056663 Q.17704 0.20909 -0.19324 169.50000 -0.20087 0.12524 0.21099 0.30473 110.0G000 ~0221344 0411990 -0-21091 -0.31527 126.00060 -0244519 -0.00358 0.19020 -6.50707 - 130.0000 0.49536 -G.09156 0.12662 0.52947 14000000 ~G.36114 -0.08965 -0.06380 -0,38943 150.00000 0.21692 --0.05350 0.02923 -0.25241 156400000 —0.15246 ‘-0.03375 -0.01819 0.19582 96 Table XX Tensor Polarization Parameters at E, = 7.479 MeV Center of Mass See Appendix E. Angle (degrees) i (T11) (T22) (T21) (T20) 17.9700U ~0.11630 -(.00731 0.02693 -0.11229 22244000 -0.13951 -09.01139 0.04161 0.13234 29.85000 -0.15565 ~0.01840 0.06367 -0.13934 37.20000 -0.16123 -9.02607 0.08322 ~0.13007 44448000 -0.14616 -0.03199 0.09322 -0.10087 51.670u0 ~0.06214 ~0.02484 0.06425 ~6.03309 58275009 0.15453 Ge018C62 -~0.05489 0.06989 65.70000 0.37921 C.98044 —-0.203972 0.10855 72.52000 0.42806 6.10823 -0.27139 0.04776 79.180900 0.37759 0.10668 -9.26701 ~0.03605 8566000 9.30778 0.09542 -0.24809 -0.11440 91.95000 0.23605 098115 -0.23140 ~0.19126 98.02005 0.15806 0.96446 ~0.21963 ~0.27486 1¢63.880090 0.06375 € 204403 ~0.21145 -0.37214 109-5060 -0.95610 C.01847 -C.20380 0248423 120.90000 -0.35986 ~G 204363 -0.17184 -0.69555 130.G0060 -9.55267 ~f.07999 -0.10414 -0.69606 14G.00000 -0.47348 -~-0.06308 ~0.04418 -0.48944 156.00009 “0.22763 ~-6.02936 .-0.00968 ~-0.24384 97 See Appendix E. Table XXI Tensor Polarization Parameters at Ey = 7.968 MeV Center of Mass Angle (degrees) i (T11) (T22) (T21) {T20) 17.97999 -0.07894 -C.0d271 C.03597 ~“0.15485 22444000 -~9.08390 -9.090335 0.04993 -0.16039 29485900 ~-O.07559 -3.00355 GO.N967T49 -0.-14135 372290690 -9.65963 -€.00281 0.07892 -0Q.10702 44.486539 -C.02636 -C.900627 C.07310 -9.05639 51.-67004% 0.06154 V.037609 C.01134 9.04255 58.75006 Cie 24484 C.01610 ~¢.15838 0.13345 65.700GU 9.40630 0.01913 -¢.32879 C.09401 72.520u9 0.242417 0.01266 -C.35738 -9.03512 79.18009 0.38276 C.09308 -0.31945 ~0.14671 85.66069 0.33822 -9.00534 -0.27799 -0.23363 92.05006 O.30967 -¢.01108 -0.25468 -0.28802 91.959900 0.29658 -0.01379 ~-C.24557 -0.31298 9B es 26C% %.25039 —Ge02321 -0.22159 -9.39773 186.900N09 0.23200 -G.02671 -C.21488 —-0242891 123.8800¢ G.18787 -0.03437 -9.29261 -0.49681 LU9.50009 C.99535 -$.04750 -0.18488 -0.61303 110.2696 .08496 —-%.204878 -C.e18318 ~0.62433 120.0UNC9 -C.21182 -3.07391 -€.13664 -~9.84675 130.6Gf04 ~-0.53348 -2.07531 -~0.06381 -0.85893 148.007 02 -~0,.55364 ~3204646 -0,.01233 -0.60217 15¢ -cegog -9.35586 -%.92119 C.05255 -0.37697 156.cC000 -~G.29803 -G6.01199 £.00388 -0.29487 98 Table XXII Tensor Polarization Parameters at E, = 8.481 MeV Center of Mass Angle (degrees) - (T22) See Appendix E. i (TIL) (T21) (T20) 17.9700 -0.04245 6.00026 0.04073 -0.17524 22244060 -6.03305 C.00110 6.05121 -0.16083 2985000 ~0.C0723 6.06362 6.05986 -0.11391 37.20000 6.02245 0.90513 0.05824 -0.05694 44.48000 (06339 6.00591 0.03112 0.02060 51-6700 6.213909 -0.09046 -0-0660C Gel2534 58-75000 0.27151 -C.02560 —6027635 0.18700 65.709090 9.38C49 -0.05925 -0.44504 0.06856 1252000 0.39574 -0.07334 0243272 -0.11549 79218006 0.37915 -~0.07475 -6.35990 -6.24359 85.66000 U.36474 ~0,.07433 -C.29625 -0.33266 91.95000 235327 -0.07534 0.24916 -0.40930 98.02000 Oe33764 —-f.07834 -C.21380 -0.49162 160.COU00 0.32964 -0.07978 ~0.20361 -0.52210 103.a60CU 0.30664 ~0.98321 -0.184460 -0.58892 11¢.000U0 (23813 0.08996 -0.15433 -0.71646 120.0N000 ~0.02545 ~(.09746 -0.09092 -0.95513 1320-00008 -G.43797 -f.C7981 -C.61096 ~1].00832 1406.000006 -0.59752 -0.04114 C.63086 -6.73496 150.0Cc000 -" 47498 -".01567 6.02957 -0.45962 156.000C0 ~(1. 36882 -6.0C B06 0.02293 -9.35717 99 Table XXIII See Appendix E. Tensor Polarization Parameters at E, = 8.971 MeV Center of Mass Angle (degrees) i (T11) (T22) (T215 (T20) 17.97000 -0.01638 C.00153 0.03924 -0.16557 22244000 %.00135 7.00259 C.04361 -5.212894 2985000 0.93547 C.00375 6.93986 ~2.05726 37.229G0G 1.C6969 ©,.00365 0.02224 0.01566 44.48000 0.1C91I1 ~%.00113 -C.02708 G.16249 51.67000 5216741 ~0.02629 ~0.15102 0.19913 58.75000 0.25915 -C.06688 -0.37333 0.21363 65.70000 7.34445 -~0.11444 -0.51730 0.02645 72252000 9237719 ~C.12395 -~C.46515 -0.19345 79.18000 O.38829 ~9.11555 -C.36322 -O.32516 85.66000 9.359915 ~£.19721 -0.28218 —-0.246625 90 .00000 Ge40727 -%9.10362 -09.24032 -~0.45293 91.95005 1.410790 ~9.10257 -0.22497 -0.47418 98.52000 %241739 ~G.10141 -0.18062 -0N.54744 103. 88000 Je41C1l2 -0.10293 ~0.14467 -0.63615 163.88000 “.41012 -~3.10293 ~0.14467 -0.63615 109.50000 . 37533 ~“CelL9697 ~Q0.11115 ~0. 74475 116.00000 7.37019 -%.10639 -9.103807 -0.75563 114.87¢600 2.29536 “0.10914 -0.07635 -9.87036 120.909099 7.15417 ~%~.1095G ~C.03852 -0.99667 130.G000G6 -0-28853 -0.08938 0.03660 ~1.10740 140.60609 -2.58113 ~0.94661 0.06451 -0.87155 150.00000 —9252340 -G.01722 0.04845 -0.57916 156.0060090 -%9.42034 -0.00859 C.03539 —-0 24493 100a Figure l. Energy level diagram for Li? with the level assign- ments of Ajzenberg-Selove and Lauritsen (1959). For the revised level assignments, see Tables VII through IX. Also see text, pp. lff. 713.325 Li’+He-a | 8.37 ( -100- Me wg (Tel) tl SOMLLLLMIAT O MAYA, —) (T=) 5.35 (te) ZAtT=07 A52., py 3,960 o+ T=I J” 174° \ 2.184 3+ T=0 4.6 |.07 L471) He*+d 10la doy 94ton'y qny Sutejoy dno Aepere gy dno Aepere gy toy yuNOW Surjetsuy LOJeNsSuUy azosseaddns oneysorjoo[y 1osseaddns syeuseypy TOF FIX Lazyoumow £9, J, "6l ST “LI “Ot “ST ‘FI "el aa | “IT 12jUNOD ZuTAouwl ayy OF sips BurutTjoqd zrojunod Areuotie;s ey} rox sit[s Sururyeq wuizojyyetd etqeisn{[py (t0}9ez302d) Tetp ostp , UoTerodz0gD UBISed Ud,, JaTUIBA TeIp OsTp ,,uoT}etOda0OD UBISEd Tq, Suttdnoo adid ,— prepueig seanjzode suturjep wereg IOJRULITIOD Wiesg Tio q souerjug sasTea yyomnys Bnqd ,,Teeg eT2175,, .¥ PevIPoW ‘33g cdd ‘yxe} seg ‘saquieyo Bulzeyeos ses oy} Jo UCTJDes~ssory “OT 6 "8 *L “9 °G °y "€e “? ‘T ‘7 oan3st a 10) SSS: MS * NS i SANS SN iI ~ rae 102a OPEC? 080 °C2 LOT °C? GGE°C7Z SSG "0c SOZ °07¢ PIP *02 £85 “07 eg7 °I2 v8 “IZ ESL “ez? GSP *92 (ooot *) sjunoy ToyTUC YT 02 Of? OST OST OST OST OST 82 09 os OF O€ (eatyezou) tosseaddng OL OL (eatitsody eZeITOA dny “ET “d ‘yxo. 99g *S1TOA 09 JO SSOOXe UT setq dATP EdeU e YIM OSeTOA Tosseaddns jo juopuedaput ATeATeTOI st UOTOeTTOD e4UL "APN 9°S 1z ,°H(P ‘P), °H Lo} ode}joA tosseaddns *sa syunoo vpTotsed peasyeog *¢ eansrg 102 (S$]0A) SOVLIOA HOSS3YddNS Ov2- O022- 002- O8!- O9!1- Ovi- OZI- OOI- O8-— OS- Ob- OZ- r | TT. TTT TT TT 18 | = 1 46 | -— — A402 = — Ae? -— 4 Ae = 4 xAve }-— — AS? -— + 49e¢ | Pop td fy SLNNOD YOLOISLAG 103a 0°VE CFL EZ? ¥°LS? S6P °ST €°SLT SET "OT 6°20T T88°S (o00T *) (Tr0= *ur5) s}UNOD peyoe1Z0yD eansselg *gt “d '}xe} 995g ‘71 = TT, Ppue APW 00°F 32 ,PH(”'*), °H worj sinsseid *sA sjunood eTayaed perayjeog +p canst 103 (}!0-W9) 3YNSS3Yd SV9 O92 Ove O22 OO? O'BI OSI Ovi OZ! OO! OS OS Ot OZ O rn rs rn re | | = + HOO! _ — 002d e = —+ HOOS jo + yoor @ , | | | Lot l SLNAOD JO YAEWNON 104a 0°0rL Ose "Sz 0°€9S G9E°6T 0°6IF G60 °FT v'SPT G8z°s (coot *) (Tfo~ sur) $}UNOD peyD.eTs105N eansseig "st ‘d ‘4x0 909 ‘2712 TPT, pue AXW 00° 32 ,o HIP ‘P), °H Wor einsseid *sa sjunod opotjsed perzeqjeog °*g eunsty 104 (1!0-W9) FYNSSAYd SVD o92 Ove O22 O02 OSI OBI Ov! OZ!I OOl O8 OF Db Oe O I | | | | iI = — HOO! = — 400d = . + 100¢e = — MOOv — + 400S | — X009 + 4 HOOL SLNNOD 40 YSEWNN 105a *umoys jou st ra}zUN0S AreuUOTyEIS OYY, “2T *d xe} 99g ‘qno Ae] Te Usut1edxe oy} jo ayeuIsyog “9g eANnsIgA 105 YAZATIVNY TANNVHO OOp 3WIL 3AIT WNYLOIdS SBWIL NIOTO YOLVYOSLNI — LNaYNNO AVNOIS 31V9 Nv 4a YSLIWONVW BD 410 YAS TdWwv “aGWYHO YSNWYVH : oy aMSIA =) wv 3a nO e eS Yal4 Id sore NOILVLS -3ud ONIdWfd 1104 [ JFONVYLNZ HOS | AATWA G3lsIGOW \ es \ } JONVL 106a Figure 7. Bias supply to moving counter from the Tennelec model #100 preamplifier. A negative bias was supplied at point C and proved insufficient for the low resistivity lithium diffused counter with the standard 22 MQ resistor in the preamplifier, so the resistor was replaced with a 2.2 MQ resistor. The moving counter with a much higher resistivity was biased through an unmodified preamplifier. See text, p. iv. 106 A 4H ° © Bo oj; oG FROM. PP OJ OF BIAS SUPPLY ‘| @ @ mag O47 pf pt 2.2MQ INPUT 047 pf FROM (oO |¢ TO FIRST STAGE COUNTER — > OF PREAMPLIFIER 107a “21 °d %1x01] eeg *soSuer yUaTIND }s0} TeyJO UT posn SCM TOJSTSAL UN GS*OOl SUL °s0}OUIOURATeS UODTGNY e YUITM f10}STSeZ UW 06866" 2U} Sssotoe doirp adejjoa ay} Aq poinseeur sem JUSTIN syy, *JOdOITSeYy MW QOT UANI QT B YIM peTseA sem pue TT?9 189} paepuejs & Aq pottddns sem Juorano oyy, *}IMOATO JUSaAND 489} 10; eAB9RUT *Qg aInst ay 107 YSLAWONOATIVS OL T (Yl) + (% 1 F) UW OSG866" Uy S800! wio > HYOLVHOSLNI OL o + 1 YOO! NHNL Ol LAOMVL = WOUS 1S3L ¢ 2O- “peo live | | { NAY 1 ff iT | | W149 sal + auvaNnvLs 108a 4 4 Figure 9A. Kinematics for He (d,djJHe , where E/E is the curve that gives the relative energy of the scattered particle in the laboratory system and E. [Ey is that for the recoil particle in the laboratory system. See text, pp. 23ff. Figure 9B. Kinematics for H-(a,0)H*, where E/E, is the curve that gives the relative energy of the scattered particle in the laboratory system and Er. [E, is that for the recoil particle in the laboratory system. Only the recoil particles were used for cross-section calculations, in that the scattered particles are either too close to the recoil particles at angles such as 19° or the spread in energy seen by the counter is too great at angles such as 30°. See text, pp. 23ff. Es /Eo, Er/Eo Es/Eo, Er/Eo 108 | | L 20 40 60 80 100 120 140 160 LABORATORY ANGLE (degrees) 180 10 20 30 40 50 60 70 80 LABORATORY ANGLE (degrees) 90 109a “Lp “dd 4x04 ay} UT TAY ANZ pe}ear} ere pue Aro}eUT]dxe-j[9S OTe VBANSIZ OY] UO saojaUTeIed Jutureulsl syy ‘sTXe Z OY} YIM °9 apsue ue }e ST YOTYM Weed JUSPTOUT 9Yy} YIM g OTSUe Ue SOYeUI UIN UL YOM Aezd possjjeos oy] WIM X ayue ue seyeul Gp eeLe po}eUTUINTII oy, ‘*sTxe Zz 94} WIOAy gq Fo vouRIsIp uUeeq @ fourtd Zx OY} UT UIeeg oY} Jo Gp JUSeUIETO Ue Wor] po.2utuiny[T duteq st pue auetd Ax oY} UL ST eeTe UOTIeJep ay, *AT}OWIOIS AOJeUITTTOO TajUNOD ey} Jo UOT]EZIIeJoUuIeIeg ‘OT eanstTy 109 110a Figure ll. Typical spectrum from d-He* elastic scattering, using a lithium drifted detector. The alpha and the deuteron peaks are labeled, where M is the number of residual counts in the alpha spectrum after background is subtracted N is the number of background counts in the alpha spectrum M' is the number of residual counts in the deuteron spectrum after background is subtracted N' is the number of background counts in the deuteron spectrum See text, p. 19. COUNTS PER CHANNEL 4000 . 3000 2000 1000 110 ALPHA PEAK 30 40 | | DEUTERON PEAK 50 CHANNEL NUMBER lila Figure 12. Simplified block diagram of phase shift analysis pro- gram. A thumbnail de scription of the program is as follows. After choosing a case out of the nine available, the values that do not vary during the iteration are calculated starting at point 1. Then cross-sections are calculated for the various angles and a ve = { (o _(N) - o (N))/ o {N) \° is calculated from the cross- sections. Next, slopes are calculated from which we can obtain changes in parameters according to Equation 13, and from these an rms value is calculated. The course the program then takes depends mainly on the rms value and secondarily on the value of ' + The rms value is compared with the previous rms value, which has been saved or is set equal to 100 radians, for the first iteration. If the rms value is larger than the previous value, the program will scan intermediate values between this set of parameters,(5,A6} and those of the previous iteration, (6', A8'), and will continue on, when the rms value is less than the previous one. The choice of 100 radians insures that the program will get past this point the first time. If the rms value meets this requirement and is less than 0.1°, the program will print its calculated values after determining polarization parameters and lolal reaction cross-section. If 0.1° < rms < 3°, the program will modify y and compare yo with the previous value after setting a control index KKW = 2. If 7 is larger than the previous value, another scanning is used until this requirement is met. If . 3° < rms < 20°, x is not checked. And in the event rms is greater than 20°, + 3° is added in the appropriate direction to those parameters whose changes are greater than + 10°. See text, p. 34 and Appendix G. All RMS 1- RMS INPUT ENERGY < ANGLES DELTAS CASE > CALC. AS Las CALC, RMS Ob -3 CALL XSECT - SUBROUTINE _» CHISQI= 20 CALC. SLOPES 8:8 +A8 [5 FOR IMAG. 8 CALCULATE CALL XSECT PARAMETERS SUBROUTINE OS CALC, SLOPES | NDEPENDENT OF DELTA FOR REAL 8 r B! 4 KKW= | RMS1=!100 2y CALL XSECT SUBROUTINE CALC. | CALCULATE || cyisge y2 AS: 5 A8 XSECT. = xX GO TO 2 , 8=38’' + AS ke—— GO TO 2 . B/ = A 8’: AS ¥ L_ KKW=e1 8:8 + AS ¥ 3 Go To z 4 CHISQI=CIQ kg 4b, = + 5° Ad)? CALC. TENSOR POLARIZATION PARAME TERS AND TOTAL REACTION XSECT { WRITE OUTPUT REACTION XSECT TENSOR POLARIZATION ENERGY, ANGLE EXP & CALC XSECT END Figure 13. Real parts of b 4! and 6 the tensor coupling parameter. (Mev) 2.935 3.441 3.946 4.450 4.731 4.955 5.473 5.961 6.476 6.965 7.479 7.968 8.481 8.971 1 § lil2a 2" phase shifts ande , See text, pp. 33ff. l ) (degrees) (decrees) 100.5 - 3.0 91.5 - 4.0 84.5 3.8 79.0 15.0 76.5 22.5 74.6 29.5 71.0 44.5 67.5 59.5 64.5 74.2 62.0 87.4 60.0 100.0 58.7 110.0 57.0 119.5 55.2 126.5 e (degrees) 14.0 ANGLE (degrees) 140 120 100 80 60 40 20 -20 ll2 L | I | | 3.0 40 INCIDENT 5.0 6.0 rae) ENERGY OF DEUTERON 8.0 (MeV) 9.0 1ll3a 3 2 1 Figure 14. Real parts of 65 > § and 8, phase shifts. The 2 dashed line is that of Galonsky and McEllistrem (1955a). See text, pp. 38ff. E 5 5° 54 (Mev) (degrees) (degrees) (degrees) 2.935 170.0 8.0 -~ 3.0 3.441 170.0 20.9 - 4.0 3.946 170.0 39.5 3.8 4.450 170.0 65.0 15.0 4.731 170.0 78.5 2205 4.955 170.90 88.0 29.5 5.473 170.0 105.2 44.5 5.961 170.0 116.5 59.5 6.476 170.0 123.6 74.2 6.965 170.0 129.1 87.4 7.479 170.0 133.5 100.0 7.968 170.0 137.0 110.0 8.481 170.0 140.2 119.5 8.971 170.0 143.0 126.5 ANGLE (degrees) 113 180 I60Fr 140/-- I20F lOoOFr 80 T 60F 4ob 20Fr 1 —20 tL. } 1 j j 3.0 40 5.0 6.0 7.0 80 INCIDENT ENERGY OF DEUTERON (MeV) 90 11l4a Figure 15. Real parts of 8 5 and 8° phase shifts. See text, pp. 38ff. E 6° 5 81 (MeV) (degrees) (degrees) (degrees) 2.935 - 7.0 - 7.0 - 7.0 3.441 - 9.1 -~12.0 - 9.1 3.946 -11.5 -14.3 -11.5 4.450 -13.5 -10.5 -13.5 4.731 -14.0 - 7.2 -14.0 4.955 -14.8 - 4.4 -14.8 5.473 -15.8 2.4 -15.8 5.96] -16.0 8.5 -16.0 6.476 -16.2 14.7 -16.2 6.965 ~16.2 21.0 ~16.2 7.479 -~16.0 27.7 -16.0 7.968 15.9 33.9 -15.9 8.481 -15.8 40.0 -15.8 8.971 -15.2 46.7 ~15.2 (A8W) NOYSLNSG 3O ADYANS LNSGIONI 06 08 02 o9 oS Ov Oe 114 q T y lJ Lf ! T Oe— lo el 28 o8 4+O¢- 09 (seasbap) FIONV Figure 16. Imaginary parts of 6 ; See text, pp. 38ff. E (Mev) 2.935 3.441 3.946 4.450 4.731 4.955 5.473 5.961 6.476 6.965 7.479 7.968 8.481 8.971 2 10.0 15.5 22.7 30.5 40.1 l1l5a ? § 2 * and 8 2 . 8.9 11.4 18.2 415 O6 (A®8W) NOYSLNSG 4O ASYANA LNIOCION| O24 09 OS Ov Of 08 T 7 t 1 so Ql- Ol O02 j OF Ov OS 09 ATONV (saaibap) ll6a Figure 17. Total reaction cross-section. See text, pp. 38ff. E o.. (Mev) (barns) 2.935 0.0 3.441 0.0 3.946 0.0 4.450 -oe 4.731 0.0137 4.955 0.0298 5.473 0.0796 5.961 0.1318 6.476 0.1879 6.965 0.2309 7.479 0.2610 7.968 0.2771 8.481 0.2867 8.971 0.2874 116 300 200 F o, (BARNS) JOO fF a. ! | | 40 3.0 60 7.0 8.0 INCIDENT ENERGY OF DEUTERON (MeV) 9.0 117a Figure 18. Differential cross-section in the center of mass vs. center of mass angle at 2.935 + 0.006 MeV. Cem. Calculated Experimental rms angle a8) o{8) error (degrees) Ymb/ sr) {mb/sr) (percent) 50.0 - 1317 21231 2.0 60.0 . 1007 - 0980 2.0 70.0 - 9857 - 0860 2.0 80.0 ~O795 -0769 Z2.U 90.0 -0790 0762 2.0 100.0 - 0819 ~0771 1.5 110.0 0868 0822 ‘1.5 120.0 ~9923 ~9898 1.5 130.0 .0978 0954 1.0 See text, p. 2landp. 46. O09 F .O8 F- DIFFERENTIAL CROSS SECTION (BARNS/STER.) 02 F Ol 117 O7 | | | a | aa | it | 40 50 60 70 80 930 100 I10 {20 CENTER OF MASS ANGLE (degrees) 130 |40 118a Figure 19. Differential cross-section in the center of mass vs. center of mass angle at 3.441 + 0.010 MeV. e.. mn. Calculated Experimental rms angle o({8) o({8) error (degrees) (mb/ sr) (mb/sr) (percent) 50.0 -1155 - 2100 2.5 60.0 0872 -0859 2.5 70.0 ~9703 20735 2.5 80.90 -0618 -0655 2.9 90.0 - 0603 - 0626 2.0 100.0 - 0649 -0629 2.0 110.0 .0749 -9734 1.5 120.0 -0896 20905 1.5 130.0 - 1080 - 1018 1.0 See text, p. 2l and p. 46. (BARNS/STER.) DIFFERENTIAL CROSS SECTION 06 fF O5-F O03 118 .09 F- .08 [- OT 04 02 Ol i | L L | | | i | 40 50 60 70 80 90 {00 I10 120 CENTER OF MASS ANGLE (degrees) 130 140 119a Figure 20. Differential cross-section in the center of mass vs. center of mass angle at 3.946 + 0.012 Mev. 8, c.m. Calculated Experimental rms angle o(6) . o({8) error (degrees) dmb/sr) (mb/sr) (percent) 50.0 . 1204 21222 2.0 60.0 .0841 . 0868 . 2.5 70.0 0602 +0653 2.5 80.0 0474 0520 2.5 90.0 0442 | » 0449 2.5 100.0 2 9500 0544 2.0 110.0 -0655 0697 1.5 120.0 0919 - 0946 1.5 130.0 - 1293 - 1288 1.0 See text, p. 2landp. 46. DIFFERENTIAL CROSS SECTION (BARNS/STER,) 119 09 F- .08 F- O07 f- .03 [- 02 -- Ot F- 1 | | a al | | | 40 50 60 70 80 90 100 Itt0 (20 CENTER OF MASS ANGLE (degrees) [30 140 L2a0a Figure 21. Differential cross-section in the center of mass vs. center of mass angle at 4.450 + 0.013 MeV. Os. im. Calculated Experimental rms angle o{®) a(9) error (degrees) (mb/ sr) {mb/sr) (percent) 50.0 » 1325 . 1410 2.0 60.0 - 0845 20879 2.5 70.0 0591 0619 2.5 80.0 - 0500 0537 2.5 90.0 - 0507 -0562 2.5 100.0 o O574 ~0615 2.0 110.0 0719 0761 1.5 120.0 -0993 . 1009 1.5 130.0 » 1443 - 1485 1.0 See text, p. 2landp. 46. DIFFERENTIAL CROSS SECTION (BARNS/STER,) O29 }- .08 F- .O7 [- .06 F- O05 -- 04 F 03 02 -- \ RUTHERFORD ~——— — | pt tp = + — =| —+ 40 50 60 70 80 90 100 HO {20 CENTER OF MASS ANGLE (degrees) 130 140 lala Figure 22. Differential cross-section in the center of mass vs. center of mass angle at 4.731 + 0.005 MeV. 8, c.m. Calculated Experimental rms Particle angle o(8) o(9) crror Observed (degrees) (mb/sr) {mb/sr) (percent) 17.97 +9067 7907 3.0 d 22.44 - 5406 ©4770 1.5 d 29.85 «3399 -3210 1.0 d 37.20 ~2415 -2260 1.0 d 44.48 1714 - 1627 1.0 d 51.67 +1200 21131 1.0 d 58.75 +0856 0814 1.5 d 65.70 - 0656 - 0661 1.5 d 72.52 0566 -0585 1.5 d 79.18 - 0547 - 0580 2.0 d 85. 66 +0565 29559 2.5 d 91.95 . 0600 -0620 2.5 d 98. 02 , -0642 0642 3.0 d 103. 88 » 0694 - 0654 4.0 d 110.0 -O774 - 0695 ' 3.0 a 120.0 2 1005 0911 2.0 a 130.0 » 1422 » 1295 1.5 a 140.0 '. 2042 ~ 1995 1.5 a 150.0 «2798 2610 1.5 156.0 23256 - 3030 3.0 See text, p. 2landp. 46. 121 1 prem 80 CENTER OF MASS ANGLE (degrees) e = eH Oo @ ~ oH x e = rey ° “4 wa ° eo e - e! ° ° i a _ { - ee a . — yo ~ _—— a e - eo eT ne ee dYOSYSHLNY L i _t i. Lo. { a i ne ro) 2 ° 2 oO fo e) 9 Q fe) O - fe) oO © S wo in ae N 7 _ _ Oo Oo Oo fe fe) Oo Oo ro) ° (YSLS/SNYVS) NOVLOSS SS08O WHLNSe aad 140 120 100 40 60 20 léza Figure 23. Differential cross-section in the center of mass vs. center of mass angle at 4.955 + 0.015 MeV. 8, c.m. Calculated Experimental rms Particle angle a(8) o{9) error Observed (degrees) ({mb/sr) (mb/ sr) (percent) 17.97 - 9020 - 800 3.0 d 22.44 - 5497 - 489 1.5 d 29.85 - 3458 322 1.0 d 37.20 . 2420 ~223 1.0 d 44.48 - 1685 -1577 1.0 d 51.67 21159 1093 1.0 d 58.75 -0818 0772 1.5 d 65.70 - 0634 0628 1.5 d 72.52 0564 ~ 0554 2.0 d 79.18 - 0566 0564 2.0 d 85.66 0601 0576 3.5 d 98.02 --- ~0691 5.0 d 110.0 -0801 -0719 2.0 a 120.0 0999 ~0922 1.0 7 130.0 1379 - 1309 1.0 a 140.0 - 1948 1857 1.5 a 150.0 ©2702 2580 1.5 @ 156.0 »3153 - 3020 3.0 a See text, p- 2l and p. 46. l2é o . e x \s He e \ +—-@— ‘ NL -- HoH ° res ° em ° go ‘ _ / — +H OO eo Z 7° oe - e oT L | aI | i | | L cam dewe | ae _ — eo) on oO fo) ro) ro) ro) Oo ra) (HS1LS/SNYV9) NOILDSS S$SOND WILEY Sisia 140 60 80 100 =. 420 CENTER OF MASS ANGLE (degrees) 40 20 l23a Figure 24. Differential cross-section in the center of mass vs. center of mass angle at 5.473 + 0.005 MeV. 8, com. Calculated Experimental rms Particle angle o(8) o(8) error Observed (degrees) (mh / sr) (mbh/sr) (percent) 17.97 - 8421 707 3.0 d 22.44 25227 449 1.5 d 29.85 -3253 » 2825 1.0 d 37.20 .2220 — 1956 1.0 d 44.48 - 1504 - 1378 1.0 d 51.67 ~1011 - 0948 1.5 d 58.75 0711 - 0679 1.5 d 65.70 - 0568 . 0540 1.5 d 72.52 - 0536 . 0490 2.0 d 79.18 .0568 ~0519 2.5 d 85.66 - 0622 -0566 2.0 d 91.95 -0675 - 0608 2.5 d 98.02 ~0715 - 0644 2.5 d 100.0 oo .0616 1.0 a 103.88 - 0747 - 0656 3.5 d 110.0 - 0786 0675 2.5 o 120.0 -0918 - 0831 1.5 o 130.90 ~ 1209 - 1075 1.5 a 140.0 1700 » L578 1.5 oe 150.0 «2340 ~ 2040 1.5 @ 156.0 2741 22520 2.5 a See text, p. 2landp. 46. DIFFERENTIAL CROSS SECTION (BARNS/STER.) 1.00 0,90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 O10 F 123 ® Va \ “~ eo. 7° Ne e-e — _- ee ~@-@-e-8 bd j { | 4 7 | ! 20 40 60 80 100 120 ~= 140 CENTER OF MASS ANGLE (degrees) l24a Figure 25. Differential cross-section in the center of mass vs. center of mass angle at 5.961 + 0.018 MeV. 8, c.m Calculated Experimental rms Particle angle a(e) o(@) error Observed (degrees) (mb/sr) (mb/sr) (percent) 17.97 7669 -6557 3.0 d 22.44 4822 -4175 1.5 d 29.85 ~ 2990 » 2680 1.0 d 37.20 . 2007 . 1887 1.0 d 44.48 1331 1258 1.0 d 51.67 0875 0861 1.0 d 58.75 0611 0595 1.5 d 65.70 ~ 9500 . 0479 2.0 d 72. 52 0495 - 0447 2.0 d 79.18 - 0545 0492 d 85. 66 - 0610 -0577 4.5 d 91.95 - 0663 -0601 3.90 d 100.0 0703 0616 3.0 fo 110.0 ~ 9730 ~ 0666 2.9 a 120.0 ~0811 0747 2.5 a 130.0 « 1043 0981 1.5 a 140.0 » 1472 1392 1.5 ce 150.0 - 2056 ~ 1907 2.0 fo 156.0 » 2428 ~ 2247 2.5 fo See text, p» 2landp. 46. (BARNS/STER.} CROSS SECTION RECTAL 1 OO } 0.90 0.80 | O10 t tt ee cement ne ns ms de a i ee ect ence oe x lO —8 eo -e-e—e—® e-—@@ al i L a zl 20 40 60 80 100 120 140 CENTER OF MASS ANGLE (degrees) 125a Figure 26. Differential cross-section in the center of mass vs. center of mass angle al 6.476 + 0.006 MeV. 8, Ce Me angle (degrees) 17.97 22.44 29.85 37.20 44.48 51.67 58.75 65.70 72.52 79.18 85. 66 91.95 98.02 100.0 103. 88 110.0 120.0 130.0 140.0 150.0 156.0 See text, p. Calculated a(9) (mb/sr) «7033 -4507 - 2798 - 1841 - 1180 - 0746 -0510 - 0431 - 0457 -0531 0610 . 0666 . 0689 - 0690 - 0687 - 0674 - 0698 0872 - 1257 ~ 1815 -~ 2180 2Zlandp. 46. Experimental (8) (mb/sr) .5930 . 4050 - 261 . 1710 .1128 .0713 0499 . 0393 .0421 . 0467 .0551 - 0590 -0625 - 0600 . 0624 0633 - 0675 .0816 . 1190 . 1663 . 1880 rms error (percent) Particle Observed aasaeenaaaAaeheweoeowma a Q oF R DIFFERENTIAL CROSS SECTION (BARNS/STER.) 0,90 0.80 0.70 0.60 0.50 0.10 Lo o.—_—_—” a 2 HH -— “e-e-0-° ee ers eo L 40 _-@-@- %e-o-—® +e—— —@ 60 80 100 120° 140 CENTER OF MASS ANGLE (degrees) 126a Figure 27. Differential cross-section in the center of mass vSe center of mass angle at 6.965 + 0.021 MeV. 8, Cem. Calculated Experimental rms Particle angle , of) o(8) error Observed (degrees) {mb/sr) (mb/sr) (percent) 17.97 - 6506 ~ 5940 3.0 d 24.44 4224 23825 1.0. d 29.85 ~2614 2505 1.0 d 37.20 - 1684 - 1614 1.0 d 44.48 . 1043 ~ 1039 1.0 d 51.67 0632 - 0634 1.5 d 58.75 0425 0411 2 0 d 65.70 -0377 -0377 2.0 d 72.52 -0432 - 0401 2.0 d “79. 18 0530 0472 2.0 d 85.66 0623 ~0557 2.5 d 91.95 - 0682 0616 2.5 d 98.02 - 0697 ~0655 3.5 d 100.0 0694 -0659 1.5 a 103.88 0677 0639 4.5 d 109. 50 - 0641 -0602 4.0 d 110.0 -0638 0621 1.5 fo 120.0 . 0603 0615 1.5 a 130.0 ~O715 - 0686 1.5 a 140.0 ~ 1045 1022 1.5 for 150.0 . 1560 - 1438 1.5 07 156.0 - 1908 1761 2.5 o See text, p- Zlandp. 46. NMTIAL CROSS SECTION (BARNS/STER.) Py DIFFER} 126 1.00 0,90 -- 0.80 |- 0.70 0.60 }- 0.50 = 0.40 |- ‘ ; x lO \ ) \, a \ ~ e eee-p—e-® -e-@ L | l 40.60 80 100 120 140 CENTER OF MASS ANGLE (degrees) l27a Figure 28. Differential cross-section in the center of mass vse center of mass angle at 7.479 + 0.007 MeV. 6, come Calculated Experimental rms Particle angle o(9) ote) error Observed (degrees) (mb/sr) (mb/sr) (percent 17.97 » 5982 ~ 5220 3.0 d 22.44 ~ 3930 - 3540 1.5 d 29.85 - 2430 - 2200 1.5 d 37.20 . «1541 . 1404 1.5 d 44.48 0927 - 0870 1.5 d 51. 67 0541 0501 1.5 d 58.75 - 0358 -0316 (2.5 d 65.70 - 0334 030] 2.5 d 72. 52 | - 0410 0367 2.5 d 79.18 0524 -0489 2.0 d 85.66 0628 -9585 2.5 d 91.95 - 0689 - 0634 205 d 98. 02 - 0700 - 0646 2.5 d 103. 88 - 0669 0634 3.5 d 109. 50 -0615 ~0618 4.5 d 120.0 0529 0540 3.0 a 130.0 0582 0507 2.5 a 140.0 . 0847 9782 1.5 @ 150.0 1305 ~ 1142 2.5 a 156.0 - 1623 » 1357 4.0 a See text, p. 2landp. 46. (BARNS/STER.) DIFFERENTIAL CROSS SECTION 127 [ 1 r T T T T b1Or _ t.00 -- — 0.90 + | 4 ry 0.80 | | | — 4 0.70 F — r ¢ ¢ } x lO 0.60 {- 4 : \ ; / : 0.50 ;- @ ry $ — 0.40 / = ; a 0.30 ;~ ° 4 0.20 + ‘\ = e e O10h \ A \ _e-e-e- e Nee “rere ee _i | a. i l i _l 20 40 60 BO 100 120 140 CENTER OF MASS ANGLE (degrees) 128a Figure 29. Differential cross-section in the center of mass vs. center of mass angle at 7.968 + 0.024 MeV. Data of Burge etal. (1952) at 8 MeV.are represented by ©. 8, c.m. Calculated Experimental rms Particle angle o{) o(8) error Observed (degrees) (mb/sr) (mb/sr) (percent) 17.97 - 5487 - 5310 3.0 d 22.44 3649 3560 1.5 d 29.85 2273 2220 1.0 d 37.20 . 1438 . 1340 1.0 d 44.48 0856 . 0787 1.0 d 51.67 0491 . 0438 1.5 d 58.75 0321 0279 2.0 d 65.70 0305 . 0266 2.0 d 72.52 0385 . 0349 2.0 d 79.18 0503 0488 2.0 d 85. 66 0609 0592 2.0 d 90.00 0658 . 0606 4.0 Py 91.95 0673 0665 1.0 d 98.02 0684 .0681 2.5 d 100.0. .0678 0656 2.0 e 103. 88 0652 0634 3.5 d 109. 50 0593 . 0608 4.5 d 110.0 0587 0590 1.5 o 120.0 0485 0482 - 2.5 a 130.0 . 0495 .0497 “15 a 140.0 0700 . 0703 1.5 0” 150.0 . 1086 . 1064 1.5 a 156.0 1362 . 1316 2.5 a See text, p. 2landp. 46. NTIAL CROSS SECTION (BARNS/STER.) ic r i . 1.00 F 0.90 0.80 0.70 + 0.60 0.50 0.40 0.20 | O.10 128 HH He ( a x O \ 08 Og -*¢ e e—6 eo. e~ e—-e~ l I | i | il l a: 20 40 60 80 (00 120 140 CENTER OF MASS ANGLE (degrees) 129a Figure 30. Differential cross-section in the center of mass vs. center of mass angle at 8.481 + 0.008 MeV. 8, c.m. Calculated Experimental rms Particle angle o(8) o(é) error Observed (degrees) {mb/sr) (mb/sr) (percent) 17.97 -4972 - 4707 3.0 d 22.44 - 3349 ~-3295 1.5 d 29.85 ~21il 2067 1.0 d 37.20 ~ 1343 21225 1.0 d 44.48 - 9800 . 0670 1.5 d 51.67 - 0457 - 0368 2.0 d 58.75 - 0295 ~0224 2.5 d 65.70 ~ 0279 -0213 3.0 d 72.52 -0355 »0325 2.5 d 79.18 0469 0443 2.5 d 85.66 - 0574 - 0570 2.5 d 91.95 - 0640 - 0656 2.5 d 98. 02 - 0656 -0652 3.5 d 100.0 .0651 0649 5.5 0 103. 88 -0629 -0649 3.5 d 110.0 - 0567 ~0528 5.5 a 120.0 0457 0423 3.5 a 130.0 » 0434 - 0402 3.0 a 140.0 ~0576 ~0574 2.0 for 150.0 .0879 .0859 3.0 o! 156.0 ~ 1102 » 1049 4.5 a! See text, p. 2landp. 46. 129 i oO N — e ——— be e _ \ - i HOH eo / re e re e wh ‘ tes °- mi °| ol e) HH e { e -H a, ea ee | GQHOSY3HLAY { i L Ll L 1 i L i L re) fe) ro) ro) ro) ro) ° Q fe) ro) oO _ ie) o © S © a ; m nN — _ ~ ro) ro) ro) fe) fe) ro) oO ro) O CYSALS/SNYVE) NOILISS SSOYD IILNSYSs3I0 100 80 CENTER OF MASS ANGLE (degrees) 140 120 40 60 20 130a Figure 31. Differential cross-section in the center of mass vs. center of mass angle at 8.971 + 0.027 MeV. 8, Cem. Calculated Experimental rms Particle angle o(8) o(8) error Observed (degrees) (mb/sr) (mb/sr) (percent) 17.97 - 4528 - 4680 3.0 d 22.44 -3100 - 3280 1.5 d 29.85 ~ 1992 » 2023 1.0 d 37.20 - 1286 . £186 1.0 d 44.48 .0776 -0614 1.0 d 51.67 0448 -0298 2.0 d 58.75 «0288 20173 2.5 d 65.70 - 0266 ~O0211 2.5 d 72.52 .0333 .0318 2.0 da 79.18 0440 0444 2.0 d 85.66 ~0541 »0580 2.0 d 90.0 .0591 .0613 3.0 d 91.95 0607 . 0663 2.0 d 98.02 .0628 - 0689 2.0 d 100.0 -=- - 0636 2.5 a 103.88 . 0607 -0681 2.5 d 103.88 - 0607 0670 2.5 d 109.5 -0557 - 0587 3.5 d 110.0 0552 - 0553 2.0 a 114, 87 0497 0493 5.5 d 120.0 0442 ~0445 2.0 og 130.0 0397 20395 2.0 a 140.0 0488 0529 2.0 fo 150.0 ~0719 .0794 2.0 a 156.0 . 0896 .0998 3.0 or See text, p. 2landp. 46. DIFFERENTIAL CROSS SECTION (BARNS/STER.) 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 130 | s 0° -% 2 ¢ _~ e e* *-e_,-* oe 20 40 60 80 100 I20 = 140 CENTER OF MASS ANGLE (degrees) l3la Figure 32. Differential cross-section in the center of mass vs. center of mass angle at 9.482 +t 0.009 Mev. 8, c.m. Experimental rms Particle angle o(8) error Observed (degrees) {mb/sr) (percent) 17.97 »4520 3.0 d 22.44 3140 1.5 d 29.85 - 1939 1.0 d 37% 20 . 1072 1.0 d 44.48 - 0557 1.5 d 51.67 ~0252 2.5 d 58.75 0140 4.0 d 65.70 0175 3.5 d 72.52 0314 2.5 d 79.18 - 0436 25. d 85.66 -0558 2.5 d 91.95 0648 2.5 d 98.02 - 0656 3.0 d 100.0 .0655 4.5 o 110.0 - 0553 4.5 oy 120.0 . 0368 3.5 @ 130.0 2 0343 2.05 a 140.0 0432 2.0 a 150.0 . 0670 2.5 a 156.0 - 0824 4.5 fo See text, p. 2landp. 46. 131 | i 1tOr t.00 + ro) ro) ro) ro) oO ro) ro) ro) (YSLS/SNYVA) NOILLOAS SSOHND WILNSdsssI1C 0.10 F- 60 80 !00 120 140 CENTER OF MASS ANGLE (degrees) 40 20 132a Figure 33. Differential cross-section in the center of mass vs. center of mass angle at 9.973 + 0.030 MeV. 8, com. Experimental rms Particle angle o(9) error — Observed (degrees) (mb/sr) {percent) 17.97 » 4420 3.U d 22.44 -3190 1.0 d 29.85 1913 1.0 d 37.20 . 1056 1.0 d 44.48 -0509 1.5 d 51.67 041] z.0 d 58.75 -0123 3.0 d 65.70 .0173 2.5 d 72.52 - 0287 2.5 d 79.18 . - 0436 2.0 d 85. 66 | - 0563 2.0 d 90.0 0649 2.5 1 91.95 0645 2.0 d 98.02 - 9708 2.0 d 100.0 . 0670 2.0 fo 103.88 ~ 0621 2.5 a 109.50 ~0554 3.5 d 110.0 -0518 3.5 a 120.0 0415 1.5 o 130.0 0329 1.5 a 140.0 0411 1.5 a 150.0 .0627 2.0 a 156.0 - 0782 3.0 for See text, p. Zlandp. 46. 132 eet ae ° e ~—— x - — em e-— \ . io e 7 — — rere 8 “a - ‘ahd . - re e et $ = i Pe eon at e — a = \ *| e bd T ——_— PY e e = tes _ \ { | _i ae i { { { j | Oo Oo © Oo (oo) oO 1) fe 8 &8 8 F & B F< BH RH F ~_ _ Oo oO oO oO ©) oO .@) Le) oO (YSLS/SNYVA) NOILOSS SSOYND WILINSS35510 60 80 100 120 140 CENTER OF MASS ANGLE (degrees) 40 20 133a Figure 34. Differential cross-section in the center of mass vs. center of mass angle at 10.474 + 0.032 MeV. 6, oem Experimental rms Particle angie — o(9) error | Observed (degrees) ‘{mb/sr) (percent) 17.97 - 4310 2.0 d 22.44 - 2900 1.5 d 29.85 - 1841 1.0 d 37.20 . 1034 1.0 .d 44.48 «0449 1.5 d 51.67 -0180 2.5 d 58.75 -0101 3.5 d 65.70 0143 3.5 d 72.52 0272 2.5 d 79.18 -0414 2.0 d 85. 66 - 0536 225 d 91.95 -0605 2.5 d 98.02 - 0639 2.5 d 100.0 9522 2.5 Q 103. 88 - 0630 3.0 d 109.50 » 0564 3.5 d 110.0 0522 3.0 a 120.0 > 0387 2.0 fo 130.0 0298 1.5 ce 140.0 0364 1.5 @ 150.0 0504 4.0 a 156.0 - 0696 3.5 a See text, p. 2landp. 46. 133 —— HH O e =, ~ x oo ry N“ a e / “ _—— e- a —_— Leroi e ro : +64 ° Nes e ~~ le ° — —™ e ma *| oe e / . oT ad 2 ane e ee e e ° e — | | l ao l L | I | | L (©) oO oO © (2) fe 8 8&8 8 & & 8B F< BH HB FE _ _ (o>) oO oO oO © © oO oO OQ (YS1LS/SNY¥VE) NOILOSS SSOY¥D WWILNS43s410 60 80 100 120 i140 CENTER OF MASS ANGLE (degrees) 40 20 134a Figure 35. Differential cross-section in the center of mass vs. center of mass angle at 10.975 + 0.033 MeV. oom. Experimental rms Particle angle a(e} error Observed (degrees) (mb/sr) (percent) 17.97 -4310 2.0 d 22.44 - 3020 1.0 d 29.85 - 1860 1.0 d 37.20 - 0946 1.0 d 44.48 - 0430 1.5 d 51.67 -0156 2.5 d 58.75 - 0086 3.5 d 65.70 -0146 3.0 d 72.52 -0273 2.5 d 79.18 -0425 2.0 d 85.66 - 0534 2.5 d 90.0 0629 3.0 a 91.95 - 0606 3.0 d 98.02 . 0648 3.0 d 100.0 0682 2.5 fo 110.0 « 0543 2.5 oO 120.0 - 0370 2.5 oO 130.0 -0271 2.5 7 140.0 -0321 2.0 a 150.0 - 0489 2.5 a 156.0 - 0648 3.5 a See text, p. Zlandp. 46. 134 “4 7 oe) —~ te nd ; = x — OL eo \ eH e 4 —_— ——— Het e a Oho % 4 \e? e — eH e ~~ — ~ iat w __ a7 r e— e 7 e e e —t | L L { { l | { [ (e) © (o) (2) © (o) _ _ Oo fo) re) re) ro) ro) ro) oO ro) (YALS/SNYVA) NCILO3S SSOND IWILNBYassI<C 60 60 100 120 140 CENTER OF MASS ANGLE (degrees) 40 20 135a Figure 36. Differential cross-section in the center of mass vs. center of mass angle at 11.475 + 0.034 MeV. 8, oem. Experimental rms Particle angle o(8) error Observed (degrees) (mb/sr) (percent) 17.97 - 4150 2.0 © d 22.44 - 3040 1.0 d 29.85 - 1823 1.0 d 37.20 . 0934 1.0 d 44, 48 0392 1.5 d 51.67 -0141 2.5 d 58.75 -0072 4.5 d 65.70 - 0140 3.5 d 72.52 0241 3.0 d 79.18 - 0397 2.5 d 85. 66 - 0509 3.0 d 91.95 - 0580 2.5 d 98.02 0590 3.5 d 100.0 - 0625 3.0 4 103. 88 0544 4.0 d 109. 50 -0531 4.5 d 110.0 0528 3.0 a 120.0 - 0347 3.0 a 130.0 - 0263 2.0 Q 140.0 - 0304 1.5 a 150.0 ; - 0443 2.5 fe 156.0 0574 3.5 oe See text, p. 2l and p. 46. 135 i | if qT | pe! i oO — I» — HOt — e ~ x ™ ia eo \ a e 7 _ te o7 _ a= =) 8 7 eH e Hey $7 Soe eo e ol e7 ~~ OL ® \ _ e@ @ a e| ee eee e Pe e 7 e e e | { | | i ! | | | | | | fo) 2g eg 8 © 8 8 gf gS Bg eG ~ — co) oO (o) (o) ‘o) ‘o) (o) oO (oe) (Y3S1S/SNYV8) NOILO3SS SSOYD IWILNSYSSIIG 60 80 {00 120 140 CENTER OF MASS ANGLE (degrees) 40 20 136a ‘97 ‘d pue tz °d ‘3x90} 90g ‘ge YAnNoOAY ZZ SOANSTY UT poisty SLOIIS VATETAL VY} JO peojsut oynposqe ay} oie peyjoyTd saeq soz19 eyT, *9¢ YSnorYy 77 saanst gy jo suot}deo 94} UT poleTNge} O1e suUOTIIeS-SSO19 OY} JO SONTeA OYT ‘WD .FT'Q + 216°LT 12 uoreynep juseprout jo ASteua *SA SseUI Jo LejUa) ey} UT UOToes-ssoao TeYUeTOTIq “peo aan3tg 136 (A®W) NOY3LNSG 30 AXYSN]A LNAGIONI oz! oll oOo! o6 or OL o9 ox Ov oe ee es ee ee es ee es ee es ee es ee ee | == —~ — ~ = oN —4 = ™ — GQYO4YSHLAN SN - — = _ ¢~— 4 ies oe = $ ; ~~ _ - _ Poy Eg YE fy | | O10 020 O¢0 Ovo ‘olen e) 03'0 Oa @) 080 (YSBLS/SNYVG) NOILOSS SSOND TIVILNSY34410 137a eSTO° 69T0° 98T0° F6TO° 9020° L020° 1v7z0° €€720* £6720° 610° PVEO* SLEO* c1v0° O€F0° TL¥0* 95950° cLS0° L€90° 0690° (-ueozed) (48 /qw)} LOLIS (e)2 - : - - - 2 * - . ° - « - - AMMA ANNNNANNN NNN NNO mmomMmnoncocdcoodownmunmMmnwomMmo oe SULIT quoultisdxy UoLeNep JuESptoUt jo Ad19ue 0£0 +986 "6 620 '+98L°6 620 "+585 °6 8Z0"+S8E "6 LZ20 +981 °6 LZ0°+86°8 920 "+482 °8 GZ0 "+485 °8 G20 "$£8E°8 P20 °tEST °8 770 °+£86°L EZO"+EBL*L €Z0°+28S °L ZZ0°+28E°L ZZ0°F18T “2 120 °+186°9 1Z0 +082 °9 610°+6LE°9 610°+6L1 9 (A°1N) Asrouq O°r €690° O° COL0° O°T €€l0° o'r 78l0° O° ¢6L0° O° £780" o°t £880° 0°2 ¢060° O° cooT’ 0°. c80T* O° ZLOT* $°T 6860° S°T soot’ Gt 4960° S°T ¢F60° GT <860° G*T 4860° S°T STOT* G°T ZeOT * ((jueored)- (1s/qut) qoaaa (e\o suit susUITIsd xy "ed *d ‘4x01 9a9 810 °+FLT'9 BI0'+8L6°S B10 °+£L6°S LIO*HELL'S 910 +215 °S 910 *#ZLE°S SLO‘HILI'S STO'+TL6'F VIO 'HOLL E10 *+89S 'F ETO *+L9E"F Z10°+991 “F Z10'+596°€ TIO °+¥9L°E 110 °#29G°€ OTO'#TIIEE 600°H6ST°€ 600 +856 "2 600°+896 °z (A°W) AZaoug "UD oF 0 + 8 °HS 12 “SA SSPUI JO 19]Ue) BY] UT UOTJIES=SsoID TenHUsTozZICG "ge oansta (A8W) NONY3LNZG 430 ADNSNA LNAGIONI Oz! Ol! ool O6 og Ou 09 os O°b oc” a SO e-— ~— - ——— . +4 100 GHOSYHSHLNY ——. = e \ —+¢oOo0d — "3 +00 * +sS00 — * 9300 = re +200 137 = as 1800 | 5 + 60°0 — ##, 4010 — | 411-0 - 21-0 pot py fy S10 (YSLS/SNYVE) NOILISS SSOHD TVILN3SY¥SS4I0 138a *d "x0} 99g 0°? 2£90° 0£0 “+986 °6 O°E L250° 6T0°+6LT °9 0°? Se90° 620 °+98L °6 0°? GLlS0° STO°+PLT'9 0°? €790° 670 °+S8S °6 G°E€ StS0° 8T0°+826°S 0°2 €290° 870 °+S8E °6 S°Z s9S0° STO*+EL6°S 0°? ¥e90° L70 °+98T "6 G°2 T8s0° LIO°+ELL*S O°¢€ 0290° L20 °+¥86 °8 G°2 0190°* 9TO°+ZLG*S O°? 9290° 920 °+P8L°8 O°E T090° 9TO°+2LE*S 0°? 9790° S70 °+S585 °8 G°Z g¢90° STO°+TLT'S G°? FL90° SZ0°+E8E°S g°e PI90° STO'+TL6°P 0°? 2290" 970 °+E8T °8 0°? 6C90° FIO°+OLL'P 0°? T590° ¥20"+€86°L 0° €lSg0° €10°+89S °P 0°? 2790° EZ0°+E8L°L G°P €TS0° ETO°+LIE'H 0°? 4290 ° €20°+78S °L G°S S9P0* cTO"+99T *V G°?¢ 9T90° 220 °+28E CL O°s 9870" Z10°+596°E G*? 1290° €20 +1882 0° Z2tg0° TLO°+P9IL'E G°? ST90° 120 °+186°9 G°gG €450° TTO°+29S °€ 0°? 1290° TZ0°+082L°9 G°P 0¢90° OLTO*+T9E*E G2 9850° 020 °+08S °9 G°G T690° 600°+6ST*E Gce S650° 610°+6LE°9 G°sS 91L0° 600°+8S6°2 (jue010d) (yus010ed) {zs /qut) Rae ak) (ajo (Ae) Ioiia (g)© (A°W) SUIL yusuttradxgy AS s0ug sua quauItrodxy ABZr0ugq ° g Aq poquweserder st *Ur*> ,[ °06 12 ( SS6T). ‘Te yo AYsuoTey jo BIVq “"Ul'D FT 'O + 56°69 128 uo TejNep JUaptout jo AZ1eue ‘SA sseUT Jo 1e}JUBD OY} UT UOT[O@S~ssOXD TETJVETETICN *6¢ ernst | 138 OI Ol (APW) NOYSLNSG 40 ADYSNZ LNAGIONI 00! 06 08 Oz 09 OS O'P O'€ | | | ; 4 Pe ee ro] 10°O c00 £00 vo0 Teme) 90°0 20°0 800 CYSLS/SNYVE) NOILOSS SSOYND TVILNAYSSSI0 139a "Iz ‘d '3x0} 90g -o¢ ySnoryy gy SeanBry Ut post] SoTL eATZTEI oy} JO P2O3sUT OINTOSGe Oy} OTE paqojd saeq 3i01z0 ay *9¢ YSnoryy gy seanBry Jo suopdes oy} ul poyetnqe; ore suoyoes-ssoL9 ayy Jo Senter OWL - @ Sq powuesordos ‘+ut'o .pT°0 + G6'T6 PLE Aq pojuesetdoa ‘euro .FT “0 40°06 32 UOLOJNEp JUSpTOUT Jo AB1SUs “SA sseUI Jo TZUN BY} UT UOTES-ssorD TeHUSTOSId “OV oInst a 139 (ABW) NOYSLINAG 3O AOYNSNS LNSOQIONI 0'O! OZ! O'll 0'6 0's O24 O'9 OosG O'P o¢e es an ae ees a a a _ — 7 t— 4 = © _ -— _ _ a & | $4 _ a 34 \ | OO OT CN 10°O c00 a OR) boo SOO 90°0 20°0 800 (YSLS/SNYUVE) NOILOAS SSOYO TIVILNAYSS410 140a "Iz ‘d ‘3x03 eeG * © Aq pajuesoides Ore "UID .OOET PUF 96 °6ZI Fe (Z296T) "12 19 FTeMOIG Jo BIEq "gE YBNO-4} QT SeaNBTA UT parsty SIOLIIO VATILTOI OY} Jo peejsur aynposqe oy} eae poyorTd sreq zozz9 eyL ‘9¢ USnorY) QT seanstg jo suotjded ay} U; peyeTNge} oie SUOTD9OS-SsOIS ay} Jo sonyeA oyy, *Ul"O ,FT°O +O‘OET 32 SUOTOND JUSpTOUT yo AS1OUa *SA SSeUI JO 19}USD Sy} UT UOTIDES-SsOAD [eTUEeTEIG ‘“[Pp oansty 140 (A®8W) NOXSLNAG 30 ASYSNS LNSGIONI O'll OO! 0'6 08 Od o9 O'S Ov O''e ! | —e—s s~_ $ ~s | 20 bO 90 80 O'| (YS3LS/SNYVE) NOILOSS SSOYD IWILNSY44si0 l4la "Tz *d "4x01 eeg “gE YSnosyI ZZ SeansTA Ur peqsty SIOLLA DATJLTII oY} JO peojsul eyntosqe oy} oae poyoTd sieq JoOAI1e ay “IE ysnory} 77 soin3st gq yO suotjdeo oy] Ul peyeTNge} ere SUOTOOS=SSOA9 SY} JO seNTeA eyT, “Ut"D FT °O + 0°9ST 12 suOLeyNep JuSpToOUT jo AFr9US *SA SseUL JO T9JUeD SY} UT UOTJIEeS-sSsOaD TeEYUSTFHIqG "7p ean3t a 14} Od! (A®W) NOYS1INAG 30 ADYSN]A LNSGIONI OO] 06 oe O02 09 Os O'bv Oe Pr ee eee eee eee ee | | +020 + 0£°0 (YslS/SNYVE) NOILOSS SSOYUD TVILNSY¥3S4410