The Elastic Scattering of Protons by Li⁷ Citation Warters, William Dennis (1953) The Elastic Scattering of Protons by Li⁷. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5XZA-X368. https://resolver.caltech.edu/CaltechETD:etd-05122003-144428 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A double focusing magnetic proton spectrometer has been so mounted as to permit observations over a continuous range of scattering angles from 0 to 160 degrees. With this instrument, the cross section for the reaction Li[superscript 7] (pp) has been measured over the proton energy range 360 - 1400 kev. Measurements were made at scattering angles of 50, 70, 89.2, 110, 130, 143.4 and 160 degrees in the center-of-mass system. Anomalous scattering was observed near 441 kev, the resonance energy for the reaction Li[superscript 7] [...], and near 1030 kev, the resonance energy for the reaction Li[superscript 7] (pp'). Analysis of the results near 441 kev indicates a state in Be[superscript 8] with J=1, even parity, formed by p-wave protons, while the analysis near 1030 kev indicates a state in Be[superscript 8] with J = 1, odd parity, formed by s-wave protons. The relative stopping cross section for protons in lithium has also been measured from 200-1300 kev. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics and Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Fowler, William A. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1953 Record Number: CaltechETD:etd-05122003-144428 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05122003-144428 DOI: 10.7907/5XZA-X368 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1742 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 13 May 2003 Last Modified: 26 May 2023 19:21 Thesis Files Preview PDF (warters_wd_1953.pdf) - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

THE ELASTIC SCATTERING OF PROTONS BY Li’ Thesis by William D, Warters In Partial Folfillment of the Requirements for the Degree of Doctor of Fhiloscphy California Institute of Technology Pasadena, California 1953 It is ¢ pleasure to acknowledge the active assistance and many helpful suggestions of Professor ’. A. Fowler in connecticn with this work. I am also indebted to Professors C. C. Lauritsen and T. Lauritsen for valuable aid and advice, and to Professor R. F. Christy and Mr. D. A. Liberman for enlightening discussions of the theoretical analysis of the data. I wish to thank Mr. E. As Milne for many hours of help in testing the apparatus and taking the data, The construction of the variable angle spectrometer mounting resulted from the joint efforts of several people. In addition to those mentioned above, Mr. V. F. Ehrgott of the Kellogg Laboratory drafting department and Mr, G. Fastle and Mr. Le Je Gilliland of the machine shop contributed greatly. I wish to express my gratitude to the National Science Foundation for a predoctoral fellowship received during the academic year 1952 = 1953, The experimental work described in this thesis was supported by the joint program of the Office of Naval Research and the Atomic Energy Commission, Above all, I am grateful to my wife Margot for her help and patience during the completion of this work, ABSTRACT A double focusing magnetic proton spectrometer has been so mounted as to permit observations over a continuous range of scattering angles from C to 160 degrees, With this instrument, the cross secticn for the reaction Li’ (pp) has been measured over the proton energy range 360 - 1400 kev. Measurements were made at scattering engles of 50, 70, 89.2, 110, 130, 143.4 and 160 degrees in the center-of-mass system, Anomalous scattering was observed near 441 kev, the resonance energy for the reaction Lil (ev %), and near 1030 kev, the resonance energy for the reaction Li! (pp'). Analysis of the results near 441 kev indicates a state in Be with J = 1, even parity, formed by p-wave protons, while the analysis near 1030 kev indicates a state in pe® with J = 1, odd parity, formed by s-wave protons, The relative stopping cross section for protons in lithium has also been measured from 200 =~ 1300 kev. PaRT ile Iii. Ve TABLE OF CONTENTS TITLE INTRODUCTION APPARATUS 1. Construction of the Variable Angle Spectrometer, 2. Methods and Uncerteinties of Calibrations. 3. Preparation of Targets. HE CONVERSION OF SCATTERING DATA INTO PHYSICAL RESULTS 1. The General Cross Section Formula. 2, Detection of Charged Particles with the Variable Angle Spectrometer, 3. Application of General Formulae to Elastic Seattering Exveriments, THE CROSS SECTION FOR THE REACTION Li’ (pp) Lai’ 1. Exverimental Resulls. 2e Discussion of Results. THE STOPPING CROSS SECTION FOR PROTONS IN LITHIUM 1. Experimental Procedure, 2. Calculations and Results. 6 APPENDIX I, CORRECTION FOR SCATTERING FROM Liv aT FORWARD ANGLES FIGURES REFERENCES P f Suh 13 13 15 37 3é 38 INTRODUCTION r; : 2 f ; . The transmitation of Li by protons has been of interest (1) for many years, in 1932 Gockroft and Walton first observed alpha particles from the reaction Lal (pa). Gamma rays from the well-known 441 kev resonance in the reaction Li! (p ¥) were observed in 1934 by Lauritsen and crane ‘”) and the width and position of the resonance were measured accurately by Hafstead and tave (2? in 19345. In 1939 creuta 4 studied the protons elastically scattered by lithium at an angle of 156° over the incident proton energy range 272 - 586 kev. He found a marked anomaly in the scattering at 440 kev and concluded that the gamma radiation from Li + p at that energy arose from a virtual level in pe”, Accurate measurements of the resonance energies of the ry reactions Li’ (p¥) and Li’ (pp') have been made by Fowler and Lauritsen’), who give 441.4 + 0,5 kev for the Li! (p ¥) resonance and 1030 + 5 kev for the Li’ (pp!) resonance, Hunt (6) has recently found 441.5 + 0,5 kev for the resonance in La’ (p ¥) using an absolute electrostatic analyzer. The angular distribution of the gamma radiation near this resonance has been determined by Devons (7) and Hine who concluded that the excited state in Bee has J = 1, odd parity, and is formed by s-wave protons. A study of the elastic scattering of protons from Li’ ean be expected to provide additional information which wil] aid in + the determination of the nature of the highly excited states of pe”, One expects to find anomalies in the scattering corresponding to the resonances in the reactions Li! (p v) and Li’ (pp!) with perhaps a small effect due to the broad resonance in the reaction ia’ (pa) at 3 Mev. The cross section for the reaction ua! {pp) has been measured by Brown, et a), at 89.2 and 143.4 degrees in the center-of=mass system, over the incident proton energy range 300 = 1300 kev. They found anomalies near 441 kev and near 1030 kev. An analysis of their results near 44] kev by cohen ‘?? indicated that the state in pe® has J = 1, even parity, and is formed by p-weve protons. This result is seen to be in disagreement with the results of Devons and Hine. This inconsistency in experimental results indicated that more complete experiments should be performed, Equipment has there- fore been constructed in the Kellogg Radiation Laboratory to study the angular distribution of charged nuclear reaction products, This thesis describes measurements of the crass seetian for the reaction La’ (pp) at center~of-mass angles of 50, 70, 89.2, 110, 130, 143.4 and 160 degrees for incident proton energies in the range 360 = 1400 kev. It is believed that sufficient data are provided to permit unique assignments for the 17,62 Mev and 18.14 Mev states in pe”, which correspond to the observed anomalies in the scattering at proton energies near 441 kev and 1030 kev. respectively. During these experiments it was found desirable to measure the stopoing cross section of protons in lithium, A relative determi- nation was performed over the energy range 200 - 1300 kev, and the results are included in this thesis. IT, APPARATUS The 1,7 Mv electrostatic generator of the Kellogg Radiation Laboratory provided the proton beam used in this experiment. The beam was rendered mono-energetic to 0.05 per cent by an 80 =~ degree electrostatic analyzer of 1 meter radius and 1 millimeter entrance and exit slits. The scattered protons were analyzed by a 1C 1/2 inch double-focusing proton spectrometer after mounting the spectrometer on a pivoted support which allows measurements to be made at any angle from 0 to 160 degrees. The protons accepted by the spectrometer were counted by the scintillation counter and pulse amplifier built by A. V. Tollestrup!'!), The general arrangement of the apparatus is shown in Figure 1A. 1, Construction of the Variable Angle Spectrometer, The variable angle spectrometer is so constructed that the entire spectrometer and the bottom half of the target chamber rotate about a vertical axis which passes through the object focal point of the spectrometer, The top half of the target chamber is connected to the entrance tube for the Incident beam from the electrostatic analyzer, and remains stationary, The movable joint between the halves is in the horizontal plane which passes through the object focal point of the spectrometer; and the vacuum seal between the halves is provided by an "O"=ring contained in the joint, The target is held at the foc2zl point of the spectrometer from the top half of the chamber. The proton beam therefore enters the chamber through the top half, strikes the target, and leaves through the bottom half. ‘his construction is lllustrateda in Figure 1B, Clearance of the movable joint is provided by directing the incident beam downward exactly 10 degrees from horizontel with the 60-degree electrostatic analyzer, Consequently, as the spectrometer is swung through 180 degrees, the accepted scattering angle varies through 160 degrees. The angle through which the spectrometer has been turned is read with an attached protractor and vernier, and the corresponding scattering angle is then calculated from the known geometry. Before the construction of this equipment, measurements were made on the gap and fringing fields of the spectrometer magnet, using a flip coil and Grassot fluxmeter. Accurate location of the object focal point was obviously necessary, and since the magnet was known to exhibit a strong fringing field it was feared that the radial position of this point might vary with field strength, The measurements were also valuable for determining the focusing characteristics of the spectrometer, Tests were made using both a brass entrance tube to the spectrometer and an iron entrance vube which was designed to shield the entering protons from the fringing field. With no shielding it was found that the fringing field was so strong thet the equilibrium path of the spectrometer was bent 11.8 degrees in the 12 inches between target and pole faces, but that the ratio of the fringing field strength to the gap field strength at the equilibrium orbit remained constant at all points over the entire operating range of the spectrometer, This indicated that the magnetic circuit of the instrument did not saturate in the pole pieces. The iron entrance tube reduced the curvature of the entering beam, but was found to become saturated and thereby to destroy the constant field ratio. The apparatus was therefore constructed with no shielding. The relative field strength along the equilibrium path for this case is shown in Figure 2, Measurements of the radial variation of the gap Pield of the magnet were also made. The criterion for double~-focusing, With axial and radial focal lengths equal, is that the field vary as rn, where n = 1/2, The results of this measurement indicete that for this spectrometer n = 0.50 + 0.02 over the range r= 10.5 + 0.8 inches (Figure 3). The focusing characteristics of the spectrometer were also measured directly with the proton beam after the construction wes completed. The spectrometer was set at zero degrees so that the incident beam passed directly into the instrument. To make the beam diverge from the object focal point, several kilovolts A.C. were applied across two parallel plates which were mounted at the target position in such a way thet the beam passed between them, The axial and radial image focal points were then determined by observing the beam at the image side of the spectrometer on a movable quartz disk, The axial image point was found to be 4 3/16 inches and the radial point 3 3/4 inches from the magnet pole face, Thus the instrument is non-astigmatic for a1] practical purposes, The separation of the focal points indicates that the field parameter n is actually 0.495. The strong fringing field of the spectrometer was found to interfere with the incident beam from the electrostatic analyzer when the spectrometer was neer the 160-degree position. The analyzer casing was therefore built of steel, and a steel lining was put inside the entrance tube to the target chamber. ‘This construction made the effect negligible. The constant proportionality of the spectrometer fringing field to the equilibrium orbit field was a great advantage as it mesnt that the flusmeter used to measure and control the field during experiments did not have to be mounted at the equilibrium orbit. It was actually mounted outside of the vacuum chamber near the edge of the gap, where its calibration has been found to remain constant over the operating range of the spectrometer, The fluxmeter used to measure the spectrometer field was built by Mr, Th. As vattne 12) | A control circuit for the Pield was alse built, which operates on an error signel from the fluxmeter. The control drives an amplidyne which is connected into the field circuit of the generator which supvlies the spectrometer. Wibh this device, the spectrometer field is held at a viven setting to within 0,05 ver cent. 2, Methods and Uncertainties of Calibrations. The 80-degree electrostatic analyzer was calibrated from observations of the gamma radiation from the resonances in the reactions Li’ (pt) at 441.5 xe ©) and Fuledl (p 3) at 993.3 key’), The kinetic energy of a varticle with charge Z traveling through the electrostatic analyzer is, non-relativistically, proportional to the potential across the analyzer plates. It is therefore given by B=6 (2.1) e Z Boot? where Foot is the reading of a potentiometer across part of the analyzer plate resistor stacks, and C. is a constant determined by the calibration. The energy of the beam when it strikes the target is less than this, however, because the barget is raised to a potential VG to suppress the escape of secondary electrons. The actual bombarding energy is therefore Ep = C. é Boot ~- Vo Le (2.2) In these experiments Vo was 300 volts; so this was a small correction. It was, however, included, The correction to Es for relativistic effects is 0.075 per cent for Ep * 1400 kev, and correspondingly less for lower energies; so it was neglected. The determination of Cp was repeated at different times during the course of these experiments, and its value was found to vary less than 0.1 per cent. The energy calibration of the magnetic spectrometer was perforned by studying the protons elastically scattered from copper, gold, aiuminum and lithium using the electrostatic analyzer calibration and the conservation laws of energy and momentum, The momentum of a narticle with charge Z traveling through the magnetic spectrometer is directly proportional to the magnetic induction B at the equilibrium orbit. The magnetic induction is measured with a moving coll, balenced torque fluxmeter; so the current flowing through the coil at balance varies inversely with B. The kinetic energy of a parbicle which passes through the spectrometer is therefore given non=-relativistically by C z° M T pot wnere Mis the mass of the particle, Tot is the reading of a potentiometer across a resistor in series with the fluxmeter coil and Cr is a constant. The ratio c/s was determined for protons by the calibration. For accuracy one must again include the effeet of the target guard potential; so that the energy which a particle passingthrough the spectrometer had when it left the target is Cy 2° 20 v 1° G pot The relativistic correction is again less than 0.075 per cent. The determination of cL /¥ was performed with the spectrometer at 90 degrees and was repeated seversl times during the course of the experiments, its valme was found to remain eenstant to within 0.2 per cent, The scattering angle accepted by the spectrometer is calculated from the spectrometer angle setting, assuming the proton beams cntering and leaving the target chamber are exactly 10 degrees from horigontal, The zero scattering angle is set by passing the direct beam from the analyzer exactly through the center of the spectrometer and settlug the protracbor veruler bo were dexreese The 10-degree beam angles were carefully checked during construction of the apparatus; so the uncertainty in the scattering angles is believed to be less than 0.1 degree. The radial exit slits of the spectrometer, which determine the resolution of the instrument, were measured with a comparator to G.0001 inches, ‘This corresponds to an uncertainty of 0.7 per cent for the smallest slit. The solid angle apertures and the counter efficiency were calibrated together by studying the protons elasti-~ cally scattered from copper, assuming pure Coulomb scattering. A previous test on copper using very small apertures measured geomet- rically had indicated that the Coulomb cross section was valid. For the calibration calculations it was necessary to know the stopping cross section for protons in copper, and the results of Warshaw \'4) were used. The stopping cross section of copper and the assumption of pure Coulomb scattering introduce a probable error of about 5 per cent in the absolute magnitude of the solid angle calibration. It was thought that perhaps fringing fields, second order focusing effects, or slight misalignments might change the spectro= meter energy or solid angle calibration with the scattering angle, or might make the solid angle calibration depend upon the size of the spectrometer exit slit, The energy calibration was therefore checked at six different scattering angles and found to remain constant within a probable error of 0.2 per cent. The solid angle calibration was reneated for each entrance aperture and exit slit combination at 90 and 160 degrees and for one combination at ten different angles between 30 and 160 degrees, The maximum probable error found for any solid angle value was 2 per cent, The current integrator capacitances were measured by J. W. Reeds to 1 per cent, The integrator firing voltage was read each day to 0.4 per cent. The resolving time of the counting system was measured, and a counting rate correction curve was drawn, The leakage rate of the integrator capacitors was measured daily, and the number of counts was also corrected for this effect. These corrections never totaled more than a few per cent. 3. Preparation of Targets. The bottom of the target chamber was fitted with an evaporating oven for making targets under vacuum, The targets o r tae SDE wal consisted of layers of natural lithium (92.5 per cent Li } deposited on a copser backing. The Layers were made very thick as the high resolution of the spectrometer gives the instrument the ability to isolate a thin lamina at any desired depth in the target material, The target was held by 2 mounting equioped with a protractor, with which the angle between the target normal and the incident beam could be set at a given value to within 0.25 degrees. The corresponding uncertainty in the scattering date is less than one per cent. The target mounting also had a vertical adjustment with ten accurately reproducible target positions. This adjustment enabled one to make full use of each target, and to compare the conditions of various parts of each target, Much Lime was spent during the course of the experiments in the attempt to obtain satisfactory targets. The requirements were a shiny mirror surface, no internal contaminaticns, and a mininum of surface layer contamination, Gontaminations were determined with the svectrometer by running a momentum spectrum or "profile" of protons scattered by the target. The target Chamber was pumped direcbly by an euxiliary sump equipped with a iicuid air trap, and another liquid air trap was installed which ran the length of the target chamber, passing about 2 centineters a . : : 7 : : pts from the target. With this arrangement vacus of about 5 x 10 “mn, Hg were mainteined in the chember during the experiments, By carefully outgassing the lithium furnace for several hours before making a : as target it was found possible to keep the cressure below 10 ~ mm. Eg even when evaporating lithium, Nevertheles: it was found necessary to deposit the lithium as rapidly as possible in order that the on oom “= target not be slightly contaminated throughout with oxygen due to continuous oxidation ef the materiel as it was deposited. by trial and error a depositing rate was determired which gave 2 target with no detectable internal contaninetion and a true mirror surface. However, after such a target was made the surface slowly oxidized even though a high vacuum was maintained; so eo be were not kept for longer than one Gay even if they were now used. Bombardment of a target with the proton beam devosited surface contamination layers of carbon and oxygen; therefore the condition of the target was checked frequently during runs and fresh targets were prevared whenever needed, Heating the target was found to oractically eliminate the carbon contamination, but it also made the lithium surface oxidize at a much greater rate; so the targets were used unheated, WWERSLCN UF SUATTIBR ING DATA LNLO PHYSICAL RESULTS 1, The General Cross Section Formule, In any nuclear reaction the total cross section for the reaction is defined as Total yield of reaction Disintegratle nuclei per unit area along incident beam vr = where the totel yielc is the number of disintegrations per incident particle, and a disintegration is broadly derined as the occurrence of the process of interest. In general, the yield will depend upon the kinetic energy of the particles in the incident beam (for our purposes we may assume the target nuclei are at rest in the laboratory system), and will have a non-isotropic distritution about the direction of the incident beam, If for each disintegration exactly one particle of type p is produced among the reaction products, we may define the differential cross section ao(E,e) . Yplbs®) = a Nt os Ke e ary ~~ where Y E98) is the yield of particle p ver unit solid angle at angle @ with the incident beam, N is the musber of disintegrable nuclei per unit volume in the target material and t is the thickness of the target along the line of the incident beam. E is the enerey of the incident particles when traversing t. Since the particles lose energy as they travel through the target material, FE is known ete > ~ 14 - only within a finite range which depends on the magnitude of t, In a detector sensitive to product p of the reaction, the number of particles recorded is n= n Ly (E58) (3.2) ey where n is the number of incident particles, f. is the solid angle subtended by the detector at the target, and 9 is the angle between the direction of the incident beam and the line of flight of the detected particles. Thus a@(E,@) 1 n(E,¢@) 1 = 2 Zz (3.3) Aan nek n Nt ° ene These quantities are all measurable, The solid angle may be determined geometrically or by calibrating on a reaction where everything else is mown, If each incident particle earries charge Ze the total number n may be determined by allowing the beam to charge a capacitance C to potential V, so n = CV/Ze. (3,4) m, is the number of counts recorded by the detector, and N is easily calculated from the chemical formula, density, and isotopic abundances of the target. This leaves the target thickness t to be determined, and it is the measurement of this quantity which usually presents the greatest uncerteinty. It should be emphasized that ne represents the number of -~1l5 = particles coming from exactly a thicimess t of the target material, since particles lose an amount of energy in traveling through the target according to the distance traveled in the target, the reaction products from a target of finite thickness are found in a finite range 4 of energy or momentum, If the detector used is selective in energy or momentum and accepts particles only over a spread § » then the detector itself determines the thickness t or the effective target if § <A. This condition enables one to use a thick target and to calculate the actual reaction thickness t from the properties of the detector, 2. Detection of Charged Particles with the Variable angle Spectro= meter. As a detector of charged particles the magnetic specto= meter is particularly suitable. It has high resolution and there- fore enables one to select a very thin target lamina from a thick layer of target material, The effective target thickness t and the bombarding energy F. are then determined as follows. Let us assume a reaction of the form A(bp)E where bombarding particle b reacts with target nucleus A producing particle p and residual nucleus B, A beam of particles b is incident with energy Ej, on the terget at angle 8, (Figure 4). The spectrometer is set to detect particles o leaving the target Pas with energy E, at angle 8, (@,,, with the incident beam), The 20 incident beam penetrates the target, losing energy, and at any point -~ 16 = a reaction may occur, If the masses of ali participating nuclei are known one carn calculate from the conservation laws of energy end momentum the quantity B/E, = 4 = a({E 8..59 Mos Myo Mi M ” ’ QUASI Soh 4 (Eqs Pape Myr Nps ‘p? ‘b) where E, is the energy of b immediately before tne reaction and is the energy of p immediately after. Bo It is necessary to introduce the stooping cross secticn e(E) = 1/N (dE/dx) where dE/dx is the energy loss of a given particle per unit length of travel through the target material and is a function of the energy EH of the parbicle, and where Ne 28 Lhe mauuber of effective stopping atoms per unit volume of the target. c{B) may be determined experimentally by direct measurements or from range-energy results. The energy of a particle b which has penetrated the target to depth x measured perpendicular to the target surface is therefore = FT tL a Ey map cos 8, Ne &,(E,) (3.5) where Ey is some intermediate energy between E, and Eyp. in Bj practice the quantity ~——>- is of the order of the resolution Hip 7 By of the spectrometer, and it can be shown thet under such conditions one may take E, = Ey in the preceding equation with negligible loss of accuracy, Therefore if a reaction takes place at depth x in the target, a product p produced =% angle Sap will heve energy -17 = xN E = ak, = obj, - 3 2 €,(E,)- me cos 8, It will again lose energy in escaping from depth x in the target, and will enter the spectrometer with energy * x . 20 ~ #2 7 Gos = N, &)(B5) aE p ~ xN, | sats; &,(E,) * we S, ca(E,) | ° (3.6) For some depth Xo» E5,' will equal the spectrometer energy setting Bo? and the particle will travel through the spectrometer and be counted, We may then solve equations 3.5 and 3.6 for Eys the reaction energy, finding YE,, +E zg, = —2_ 22 (3.7) 1 a +” cos 8, €,,(aB, ) where : (3.8) ~~ ' cos 8, e,(E,) In practice the target is usually set so that 8, = 855 making cos 8, /cos 8, = 1, To find the target thickness selected by the spectrometer we differentiate 3.6, with EB! = Boos finding dx N. WBog 7 Sos 8, €, (8, )(a +74). If |az,,| is equal te $2 the energy spread accepted by the 20° spectrometer, then the target thickness selected is just dx/cos 815 - 18 = so that N > = 35, (a +9) €,(E,)- (3.9) o We can express this in terms of the resolution calculated from the width of the detector window, which for a magnetic spectrometer is given by P E = 2 SP SE, pa] 1 Yr 2) 2(1 + M) os ea i where M is the magnification, Py the equilibrium orbit radius and Sr the radial width of the exit slit of the instrument. In terms of the resolution, equation 3.9 becomes 1. Ez O° ae MoCo +7) ©, (8) ry (a+) =) EE ET Ne ult) (3.10) Introducing equations 3.10 and 3.4 into equation 3.3, our expression for the differential cross section becomes yea ao s Ze R an (89 -F QV Se Sy fe) oy (8) 9 98) (3.11) - 19 = where f | /2 f { Mf Me Ey (Me) Me Y M, - My qt cos 8 ap + a a ~ 5 wth fo o#} v + E, My + My Ma M B E, Mp Mp Ma » 1/2) * + t—- cos g (3.12) 2 LAB (Mp + My) and — _ 2 Q = (Meo * Mao ™ Mo Mpa) Cs the subscript o denoting rest mass plus mass equivalent of any excitation, Equation 3.11 is valid in either the laboratory or center-of-mass system, depending on which system is used to express the quantities M and @. 3, Application of General Formulae to Elastic Scattering Fxperiments. In applying the above general formulae to the experiments on the elastic scattering of protons from Lil described in this thesis, the following simplifications were made, The target and residual nuclei are identical in an elastic scattering vhenomenon, as are the bombarding and emitted particles b and p3; so in this case 2 = 0, and equation 3,12 reduces to te Mi 2 1 1 = = [—*- cos @ + ar ~m* sin” @, 1/242 1 E mt+tM LAB mtM LAB (3.128) where m is the mass of the proton and M the mass of the scattering -~ 20 - nucleus. Yor any given elastic scattering process, therefore, a depends only upon the scattering angle. The depth in the target of the actual target lamina from which data are taken is of some importance. If the lamina is too deep, straggling becomes very large and the energy resolution of the data is destroyed. If the lamine is too near the surface the exit window of the spectrometer may not be filled, rendering equation 3,11 invalid. The compromise foliowed in this experiment was to operate with the lamine beneath the surface a distance which was a eonstent small multiple of its own thieknegs. This as the proton bombarding energy was varied over the investigated range, the setting of the spectrometer was also varied in such a way as to fulfill this eriterion. For such a "following curve", using equations 3,6 and 3.9 for the depth in the target x/cos 6, and the lamina thickness t, the criterion 5 = nt gives wa 1 § a OBB ~ E56 ~ 2 B20 “ne (3.13) or Eo 0. R Introducing the constants of the apnaratus from equations 2.2 and 2.4 and ignoring the small target potential corrections we have oO Cy Bot Toot a a+ R ) so that RO zg = Ky Zoot Ioot * E K is designated the "following constant", and depends upon the scattering angle, the constants of the apparatus, the resolution setting of the spectrometer, and the "following depth" n in the target. In this experiment K was held fixed for each angle over the energy range studied, and n was always near 4. Calculation of Ey from the experimental settings of E 1B and E55 by use of equation 3.7 is complicated by the fact that the quentity % is itself a function of E,, However, the variation of By with 9 is smeil, ac is show by differontiating equation 3.7, substituting 3.13 and using the definition of e(E): dE, jy 78, 2 $(,) am (atn)y a ey where n is the "following depth" of the target and § {E,) is the energy thickness of the target lamina, Por these experiments n was about 4 and (a +?) near 2; so da, ~ 2dy j (E,)- Therefore one needed to snow ¥ only within approximately 0.5 in order to determine Ey within the uncertainty introduced by the finite target thickness alone, If one assumes an expression for c{Z) of the form mY on Cc where the consbants C, a and b are determined empir curve for e{£), then with 8, = 6, equation 3.6 beex celly from a ome 8 \ 4 atbe ey . ef) = th about some value ES we obtain 3 a od Sencoat Hy ! md ete : ca) ry as + Thus the error in %(&) introduced by usi using ¥% = % = -. toe . w- 1,07" , is AY () = ) ° be - EB) log - For lithium, the stopping aa eross section e(©) is ery y Lye dy co SMPLPica well represented by the ebove formula over the energy range 200 = 1500 kev if one takes a and b= 0,00012. Since the present exoeriment the energy range 406 ~ 1400 kev, we take Sind bs were performed over BE * 900 kev, which gives 1,0.71 io Tee, ANE) < 0.05 over the entire range of angles and energy studied. This is seen LO cause an error in the determination of £ =, dess than ore tenth the energy thic! kness of the target lamina, which is negligible. % _ was therefore used throughout this work in the calculation f IV. TEE CROSS SECTION FOR THE REACTION Li! (; pp)Li. wed & Bxperimental Results. The elastic seattering of protons by lithium was studied ttering angles of 50, 70, 89.2, 112, 130, 143.4 and 160 go af, se degrees in the center=enfemass system over the incident proton energy range 360 = 14CC kev, The ratio of the observed cross section to the Rutherford seattering cross section over the proton energy range 360 = 500 kev at the angles studied is shown in Figures 5, 6 and 7, The center-of-mass differential cross section over the energy renge 500 - 1400 kev for these angles is shown in Figures 8, 9 ard 10. Tables 1-7 give the experimental results for the differential cross section and for the ratio to the Rutherford scattering at each energy measured for the various angles. The maxima in the scattering for proton energies near the 4é1e5 kev Li’ (p ¥) and the 1030 kev Lai’ (pp) resonances are shown in Table 8, The values given in Table 8 include corrections for the resolution effects of the apparatus, while the others do not. The sources of error and uncertainty in these measurements have been discussed in detail in the preceding sections of this thesis. The total uncertainty in the energy calibration of the apparatus is below 0.5 per cent, and non-linearities and systematic errors are believed to be less than 1 per cent. The orobable error n the absolute magnitudes of the various spectrometer solid angle apertures is of the order of 5 per cent, while the uncertainty ~ 2h = between the relative values of different combinations of solid angle aperture, spectrometer exit slit and current integrator capacitance is another 2 ver cent. The reproducibility of results, which is affected by errors in settings, target smoothness and contamination, straggling, et cetera, was within 5 per cent, Another 5 ver cent uncertainty in the results is introduced by the stopping cross section of protons in lithium, Taking the root mean square of the contributing uncerteinties, one finds that measurements is of the order of 10 per cent, and in the relative magnitudes is of the order of 5 per cent. TaAshé 7 ° in barns /steradian and ratio proton energies in kev. Observed differential cross section ta Rutherford cross seetion for ineident E T/r, aeya n E o/e., dg/an oe /, ag/an. 355 1,12 4e2l 458 1,17 2605 968 oO7 od 365 1.11 3.95 460 1,16 2.01 978 290 045 395 1.13 3,89 402 1.74 2.48 ORR era A697 385 1,11 3.56 464 1.12 248 998 099 ATL 39 1210 3.35 468 1.75 24650 1007 1,01 476 399 1,09 3425 4AT2 1,13 2e42 1017 1.04 480 403 1.07 3613 A476 1,12 2035 1027 1,08 0485 407 1.06 3.05 480 1.13 2034 19037 1.11 09h 411 1,07 3,01 ABL 1.13 2.30 1047 1.14 0495 AV5 1.05 2290 L488 1.12 22h, 1057 1.13 0485 49 1.05 2685 492 1.11 2.18 1067 1.12 oA72 423 1604 276 494 1.10 2.15 1077 1.11 0455 427 1.92 2.06 514 1.09 1.97 1087 1.08 fd 429 099 2.56 533 1.09 1,83 1096 1.09 43 431 1.00 2656 553 1.08 1,68 1106 1.07 42 432 099 2053 573 1.07 1.56 1116 1.06 Al A3L 1,01 2650 593 1,08 1.46 1126 1.04 039 43 1.02 2.50 612 1.07 1.36 1136 1.03 038 £38 7 Aa) 2.83 632 1 005 1 025 11 46 1 002 37 439 04 1.08 2.66 652 1.05 1.16 1156 098 035 44004, 1.11 2672 672 1.03 1.0 1166 1,00 03) 441.3 1.13 2.76 691 1,02 1.02 1176 096 033 44263 1.18 2.86 711 1,02 296 1185 096 032 443 03 1,19 2089 731 1.01 090 1205 oh ol 44503 1.23 2095 770 97 78 1245 090 028 4463 1.25 2.98 790 096 oT 3 1264, 087 025 447 3 1.24 2.95 810 095 2&9 1284 Oa 2h, 11843 1.24, 2.93 $30 92 064, 1304 80 223 449.2 1.22 2,88 849 90 259 1324 075 e2il 450.2 1,21 2684 869 08h 056 1343 076 2200 451.2 1,21 2.83 RRO 86 ~52 1363 7h ~189 45242 1,21 2,81 SO9 0 fh 48 1383 073 184 456 1.18 2670 QE 034 045 wo DE = TABLE Z Li Ton), @ eu = 70°, Observed differential cross section in barns /steradian and ratio to Rutherford cross section for incident proton energies in kev. oo o/a, ag/aAn Ep o/¢. ag/an ED c/o, agen B54 1.06 1,19 449 1.34 293 671 92 ae 364 1.0 1.12 4,50 1.32 291 691 093 oad 374 1,04 1.04 451 1.29 089 710 29C 025 384 1404 9 452 1,428 Paolo) 730 oO 0 2h 394 1.03 093 453 1.27 087 750 236 “2th 402 1403 239 455 1,22 082 T89 24 iE 404 1,03 289 456 71420 281 809 285 0182 4,06 1.04 089 L457 1.20 080 829 682 2167 408 1.04 08S 458 1.19 080 849 281 0158 410 1,03 286 459 1.18 +79 868 282 0153 412 1,03 085 1,60 1.18 07S 888 083 0147 Al 1,02 Sh, 461 raw) 078 9Os 085 ILA 416 1.04 084 £62 1.17 of? 928 91 0148 418 1,02 082 463 1.14 075 947 7.00 0156 420 1.03 82 464 1el4 oth 967 1.16 WPA 42) 1.02 081 465 1.14 07h 987 1.30 2187 Ape 1,02 80 4,66 1.14 073 997 1.39 6196 423 1,03 oS 468 1ei2 vf2 1007 1.47 on 0d 42k, 1,02 080 470 1,70 70 1016 1.50 2204 1.25 1,01 078 472 1.12 070 1026 1,51 0201 426 1,03 280 4th 1.09 068 1036 1.53 2200 427 1,03 279 476 1.09 067 1046 1.53 196 428 1,03 of9 478 1.10 207 1056 1.51 190 4z9 1.03 th) 480 1.11 067 1066 1A? 2181 430 1.04 079 482 1.09 266 1076 1.41 0171 4,31 1.05 379 L484 1.12 207 1085 1.37 163 132 1.04 078 436 1,09 065 1105 1.29 0143 433 1.07 80 L88 1.08 eva 1125 1,21 0134 434 1,08 220 490 1,30 264 1145 1.14 122 435 1,09 Si 492 1,09 03 1164 1,07 2110 436 1,13 083 493 1.08 262 1184 1,04 2104 437 1,15 «84 494 208 261 1204, 098 2094 4.38 i,ig 236 496 208 261 1224 296 2090 439 1,23 289 498 i 013 264 1243 096 2087 £40 1,30 294, 500 1.11 262 1263 91 2080 L4| 1.36 298 513 1405 256 1283 090 2C76 442 1 he 1 202 533 1 2 Dd, ® 51 1 303 28 2072 443 1.45 1,03 553 1,04 48 1322 ol 2073 4AL 1.47 1.04 572 1,00 oe) 1342 87 2007 445 1.48 1,04 592 098 239 1362 290 2068 4d6 1,46 +03 612 1.00 037 1382 ey 2067 447 1.42 99 632 7 a 34, 448 1.40 298 657 296 032 = 27 = PABLE 3 Li’ (pp), 8.4, = 89.2°, Observed differential cress section in wee AE, barns/steradian and ratio to Rutherford cross section for incident proton energies in kev, Ey /o,, ag/an. Ey o/r, av/an E. v/r, av/an 358 1.03 250 508 = 96 023 1016 2.612 4,128 363 1.03 048 518 095 222 1026 2.11 6126 378 1,01 ahd, 538 092 0199 | 1036 2.09 9122 388 97 o4C 558 290 .181 | 1046 2.03 .116 398 1.00 239 58 238 0165 | 1056 1.98 ,111 4,00 1,00 039 598 237 2152 | 1076 1.77 096 408 99 oo? 617 285 2139 1096 1,61 084 £18 1,02 0365 637 233 2128 | 1115 1.48 075 128 1,08 037 657 283 2120 | 1135 1.39 -.067 433 1,23 41 677 .80 2109 | 1155 1.29 060 434 1.28 043 697 279 2102 | 1175 1.21 055 435 1.32 ohh 717 76 2092 | 1195 1.17 2051 4,36 1,38 45 737 077 2088 | 1215 1.14 6048 A437 1.46 048 157 075 2082 | 1235 1.12 O46 £38 1.56 51 777 075 e078 | 1255 1.12 044 439 1.63 053 797 07h 2073 | 1275 1.13 043 440 1,66 0535 &17 0/5 0070 $ 1295 1.12 042 441 1.70 0545 837 077 0069 | 1315 1.14 041 442 1.68 0537 857 280 2068 | 1335 1.18 2041 £43 1,66 053 876 Asie) 0070 | 1354 1.23 0042 LAL 1.61 abl 896 093 2072 | 1374 1,25 0041 LL8 1.37 A3 916 1.05 2078 | 1394 1.27) 041 458 1.08 032 936 1,23 2088 468 1.01 029 956 1.44 2098 LTB .97 227 976 1.74 .14, 488 296 025 996 2.00 0126 498 296 ek 1006 2.10 0130 - 28 - TABLE 4 Lal (pp), Son = 110°, Observed differential cross section in burnus/steradian and ratio bo Rutherford cross secbion for incident proton energies in kev. E oF, ao/an. By Oe, ao/an. E., o/o, ar/an 359 099 026 44h 1.64 028 817 075 2038 369 97 02k 445 1,54 026 837 82 2939 379 097 223 446 1.45 025 857 093 ~043 388 698 022 147 1.34 2225 877 1.03 2045 398 98 ofl L448 1.27 213 897 1.19 2050 400 98 0205 450 1.15 6191 917 1.42 2057 402 1.00 2208 4.52 1.02 2168 937 1.76 -068 404, 099 2206 L454 1.00 0154 957 2.16 2079 406 1.01 0206 456 095 2153 977 2258 2091 £08 099 2200 458 095 2153 987 2679 096 4190 296 192 462 091 e144 997 3,03 2102 412 1.00 0198 1,66 289 2139 «| 1007 3.15 2105 AVA 1.02 © 200 470 289 .137 «=| 1017 3.21 0104 416 1.01 0196 47h 085 2128 =| 1026 3.19 .102 418 1.02 0198 L78 8h .124 | 1036 3.11 2098 420 1.03 -196 482 288 .128 11046 3.01 .093 422 1.06 .200 486 285 6121 | 1056 2.93 .088 424, 1.09 0203 490 yA 6117 | 1076 2.63 2077 4,26 1.16 02ld5 494 84 0115 | 1096 2.37 »067 128 1.15 21 498 285 .112 | 1116 2.17 2059 429 1.21 0221 518 083 104 | 1136 2.00 052 430 1,22 022 538 681 094 |1156 1.91 048 431 1.26 023 558 “78 6085 «| 1176 1.84 2045 432 1.31 02h, 578 76 077 1196 1.81 04,3 433 1.37 025 598 27h 2070 | 1216 1.78 041 A34 1.44 260 518 73 2064 | 1236 1.76 2039 435 1.49 onl 638 273 -060 | 1256 1.78 038 436 1.59 228 658 69 054 | 1276 1.87 039 437 1.67 229 678 269 2051 «| 1296 1.90 2038 438 1.74 230 698 267 2046 11316 1.97 038 439 1.85 032 717 07 e044 | 1335 2.03 .038 440 1.87 2325 737 268 2042 | 1355 2,12 6039 LAA 1.84 032 757 269 2041 =| 1375 2,21 039 £42 1.82 31 777 «69 -039 | 1395 2.31 O40 443 1.72 «29 797 273 2039 - 29 — TABLE 5 ai’ (pp) , Oa = 130°. Observed differential cross section in barns/steradian and ratio to Rutherford cross section for incident proton energies in kev. E. o/t, da/an B 0/0, do/an. E o/0-, dav/an 359 09, 160 469 TT 2079 960 2.92 071 369 .97 2160 479 -73 2072 970 3,28 .079 379 095 0149 489 272 068 980 3.64 2085 389 296 143 499 75 2068 990 3496 2091 399 095 0134 500 276 08 1000 4025 2096 4,03 -o7 0134, 520 277 064 1010 4.48 2099 L407 098 2132 540 276 2059 1020 4.66 2101 411 96 0128 560 7h, 053 1030 459 .097 415 299 2129 580 072 048 1040 ede 2094 419 1,03 «132 600 269 043 1050 4239 2 O90 423 1.07 0135 620 69 -040 1060 1022 084 127 1.18 2146 640 266 2036 1070 4,00 -079 429 1.26 0154, 660 265 034 1080 3.92 .077 431 1.31 2158 680 66 032 1090 3.70 070 433 1245 17h 700 265 030 1100 3.59 067 435 1.61 2192 720 66 029 1120 3.32 060 £37 1.85 2218 740 267 2028 1140 3.15 2055 1,38 1.92 3226 760 668 027 1160 3.02 -050 439.3 1.97 -230 780 072 2027 1180 2.93 047 £4043 1.99 0231 800 277 027 1200 2.92 2046 141.3 1.95 0225 820 285 .028 1220 2.88 043 142.3 1.92 2220 840 92 2029 1240 2.91 6043 £4303 1.75 -200 860 1,06 032 1260 2.93 e042 LL5 1.49 2169 880 1.23 036 1280 3.03 2042 LL7 1,22 2137 900 1.46 2041 1300 3.10 2041 LL9 1.12 0125 910 1.63 oOL4 1320 3.21 2041 453 91 2100 920 1,83 0049 1340 3.32 2042 457 80 086 930 2205 2053 1360 3.41 2042 461 282 087 940 2034 060 1380 3.56 «O42 465 79 .082 950 2.63 066 1400 3.65 2042 = 30- TABLE 6 ia? (PP) > Con * 1430406 Observed differential cross section in barns/steradian and ratio to Rutherford cross section for incident proton energies in kev. By o/r, ar/an, Ey ole, ao/an. E o/G,, ag/an 365 99 139 £48 1.15 2107 618 258 2029 317 099 2130 449 1el4 0106 638 257 2026 389 296 0119 450 1.13 104 657 257 2025 395 297 2116 451 291 2084, 677 057 2023 401 298 RPA 452 90 6082 697 059 2022 403 095 .109 453 89 =. 081 717 359 0214 405 99 2113 4,54, 90.082 737 -60 0206 407 97 2109 455 289.080 757 262 0202 4.09 1.00 2112 456 92 =. 083 7177 207 0208 411 1.00 2110 457 94 = 6084 797 +73 20215 413 1.02 2112 458 87 =. 078 817 -79 .022 415 1.06 0115 459 79 2070 $37 27 2023 417 1.16 0124 460 90 36079 857 1,11 2028 419 1.04 2110 461 83 2073 877 1.17 2029 421 1.08 0114 163 .78 =, 068 897 1.47 034 422 1,01 0106 465 085 =, 073 917 1.86 2041 423 1.25 2130 467 76 2065 937 2.36 2050 A2k 1.15 0119 479 81.069 956 3.10 064 425 1.17 0121 471 «73 061 976 3.85 2075 426 1.21 0124 473 «77. O64 996 Ae57 2086 427 1.16 2119 1:75 77 064 1016 5.04 2091 428 1.16 0118 477 .83 =, 068 1036 5.12 .089 429 1.36 0138 479 75 061 1056 485 .081 430 1,31 2132 481 76,061 1076 LeAT 2072 431 1.34 2135 483 75 =, 060 1096 4el4 2064 432 1.39 139 485 73° =, 058- «| «1116 3.91 2059 433 1.46 0145 L487 077 2061 1136 3.72 054 434, 1.50 el 49 489 °76 260 1156 3.54 2049 436 1.74 01771 493 76 4059 1196 3.47 2045 437 1.86 182 495 281 062 1216 3454 2045 438 1.97 0192 497 -79° 060 1236 3.53 04,3 439 2.08 2202 498 266 =. 050 1255 3.63 2043 440 2612 2205 499 78 2058 1275 3.71 2043 LAA 2.12 2204 501 82) 061 1295 3.79 2042 LA2 2.02 0193 518 66 046 1315 3.87 2042 Lhd 1.73 164 538 oh 041 1335 3.97 2042 445 1.58 0149 558 263 038 1355 413 2042 446 1,41 0132 578 61) 6034 1375 4.20 2041 4L7 1429 2121 598 058 030 1395 441 2 04,2 proton energies in kev. barns /steradian and ratio to Rutherford cross section for incident M = 160°, - 31 - TABLE 7 Observed differential cross section in E o/T, dc/an. E c/o, advan E, o/C., ar/an. 373 1.00 2116 471 287,063 866 1.67 034 377 99 2112 476 076 054 $98 1.91 2038 381 1,01 2112 481 e280 =, 056 908 2.10 6041 385 1,02 2117 485 +77 053 918 2.30 2 O44, 389 1,00 2107 487 273 ~=—-050 928 2062 2049 393 1,03 107 489 -78 6053 938 3.03 0056 397 99 2101 491 281 0054 948 3655 2064 401 1.02 .102 493 76 =, 050 958 3.94, «069 405 1.03 2101 495 ~76 = -.050 968 4036 2075 409 1,05 .101 497 «776050 978 4eIk 2083 413 1.05 2099 499 277 050 988 5.56 0092 417 1.11 2103 501 276 049 998 5.98 097 419 1,12 2103 503 277) ~— 049s |: 1008 6.30 2100 421 1.19 2108 519 074 =e OAL «| 1018 6.70 0104 423 1.16 0104 539 e71 = 040 =| 1028 6.79 2104 425 1.20 2107 559 -70 =, 036 |: 1038 6.88 210% 427 1.24 2110 579 68 =.033 | 1048 6.74 0099 429 1.37 2120 599 66 =.030 | 1058 6.52 0094 431 1.47 2126 619 068 = =,029_-| 1078 6.15 2085 433 1,62 2139 639 060 = 024 «|: 1098 5.64 076 435 1.81 0154 659 061 = 0223 | 1118 5.40 .070 437 2.03 0171 679 264 0224 | 1138 5.19 2065 438 2.12 178 699 062 = 0205 | 1158 5.01 -060 439 2.19 2183 719 067 = 60210 | 1178 4.99 2058 140 2626 2188 739 266 = =.0195 | 1198 4.87 2055 441 221 2183 759 70 =°,0197 | 1218 4.76 2052 142 2.10 0174 779 «71 = .0189 | 1238 489 2051 £43 1.95 2160 799 86 = .0218 | 1258 4.95 ~050 AA5 1,69 138 809 90 = ,0222 | 1278 4.96 O49 AL? 1,33 .107 819 992 =. 0221 | 1298 5.20 »050 LL9 1.12 .090 829 96 =, 0226 | 1318 525 2049 151 1,00 079 $39 1.10 .025 | 1338 5,38 048 453 093 073 849 1.12 2025 | 1358 540 2047 4,56 688 .068 858 1.23 027 | 1378 5.60 2048 461 cy 061 868 1.39 .030 | 1398 5.70 0047 166 084 2062 878 1.47 031 Maxima of the observed differential cross section and of the ratio to Rutherford cross section near the 4/1 and 103BC kev anomalies at various center-of-mass angles. These values have been corrected for the effects of resolution in the accaratus. Jlote that at angles greater than 100 degress the results near the 1030 kev anomaly indicate spherical symmetry. “owt * 50° 7089.29 110% 130 143.4% 160° 9/0 Wise, 1449 1.70 1.82 2,00 2.16 2028 kev ay /inl} 2.97 1.04 2 5L9 6328 5233 6207 £190 of, 5 ” o “ aon , 1030 Me \l ag 1453 2012 3.21 4064 5618 688 % eV ar/an 49 1204 4129 (105.101 ol “04 The cross section for elastic scattering of protons by ~Li nas previously been studied by Trown, Snyder, Fowler and Laurillsen at &9.2 and 143.4 degrees in the center-of-mass 2 3 Toe (15) 4 or 3 YL. 3 system, and by Bashkin anc Richards at 165,23 derrees, The work resent o of Prown, et al., was done in the same laboratory es was the ¢ work, The two results at 143.4 degrees are brought within 6 ver cent agreement over the entire energy range studied if Prown!s data are recalculated using the curve oe bee ea which was measured and used in the present work, However, thi correction leaves Brown's results at €9,2 decrees up to 20 per cent < lower than the present results, Because of improved experimental techniques, the present results must be assumed to be the better, Eashkin and Richards give ~ 0.08 barn/steradien for the maximin ” differential cross section near the 1030 kev anomaly at 166,23 degrecs, 48 their uncertainty is 20 per cent, the agreement with the present work is within experimental error, A theoretical analysis of the reaction Li’ (pe) is complicated rn by the fact that the ground state of Li’ has non-zere anguler momentum, This fact also means that, exceot ir the case where the relative “ angler momentim Q of the affective incident proton vaves ig zero, a coupling scheme by which one can arrive at a given totel momentum state of the compound nucleus is not unique. ince the engulur distribution of the scattered protons depends on bhe Lype of = 3L, = coupling, one expects an analysis of experimental data to vrovide information short the coupling of aneuler momentum vectors in the nucleus. If one specifies the type of coupling exectly, as Russell- ctunders coupling, a definite anguler distribution of the reaction products is expected. The question of coupling may be sidestepped by introducing the chennel 4 s of the incident particle and the total anguler momentum I of the target nucleus. Por an incident particle with s = 1//%, S has twu values, [I -s, and 1+ s. The contributions of the two channels to the formation of a given compound state of total anguler momentun J are independent, and the theoreticel angular distribution of the reaction products may be expressed in terms of an additional parameter, spin S, which is the vector sum of the he the mixing ratio t, which represents the ratio of the contribution of channel spin 5, bu thal of channel spin S,. This parameter then Ke eveluated experimentally. 7 The angular momentuis of the ground state of Li known to 3/2, and its parity is generally assumed to be odd, ‘The spin of the proton is 1/2, and parity is taken as even, The savyeilable channel spins are therefore S = 1, 2, and the channel voarities are = 0 can odd. Incident proton waves with J = S, odd parity, and there is no arbitrariness scheme. However, incident croton waves with Q= 1 can states with J O from channel spin 1, states with J wates with J = from channel Ss from both channels, anc with even parity. ‘the nonexistence of alpha particles form only compound states : in +} the couplings form nd compou corresponding : 4 4 ef , to the 441 kev Li (p 3) resonance enables one to ascume that the ¢ related state in Be”~ does not heve hoth even anpuiar momentam and aven parity, consequently we need be concerned only with the possibility that this stete has J = 1 or 3, if itis formed by p-wave orotons. A preliminary theoretics] analysis of the present data hes om “Sf o °, > a he is es been performed, using the single-level formulation given Some of che theoretical curves for the angular variation of the maximum ratio of elastic seattering cres: section to Rutrerford crass seetion near 441 kev are plotted in Figure 12, with the exverimentel values included for reference, The best fit to the data is obtained by 4, e formed by c-wave protons, ae assuming 2 7 = 1, even parity, state in oi gith a mixing ratio t = 1/2 between channel spin 1 and channel spin 2. In this case the theoretical angular distributions of the ry cd o = © associated gamma rays ere 1 + C.1€ cos” @ for the 17.6 Mev radiation 2 a a . ge 4 mL. Oth and 1 + 0,015 cos” 6 for the 14.8 Mev radiation, The fit if one ssumes a mixing ratio t = 1/5, which corresponds to the "LS" assign- 3 nent 34 and woich is necessary to explain the isotropy of the ganma toe s a,t7) , . . rediation found by Devons and Hine ; is not quite so good; but this possibility is certainly not excluded. A more rigorous theoretical s+ treatment would undoubtedly be advantageous. However, it seems clear pee 8 that the vossibility that the state is formed by sewave orotons definitely excluded. Hear 1030 kev, the differential cross section resuits for scattering angles greater then 100 degrees are consistent with spherical symmetry, which indicates that the reaction is induced by s-= wave orotons, Singic-level theoretical curves for the engular variation of the chanve in magnitude of the ratio of the cross section to Rutherford near 1930 kev, assuming Q= 0, are plotted in The fit toa the data of the eurve for J = 1 is good qualitatively, and within about 25 per cent quantitatively. The effects cf broad resonances above 1 Mev and the influence of competing reactions, both of which are neglected in the present theoretical treatment, can easily be blamed for the lack of quantitative fit. Again a more rigorous theoretical treatment would be advantageous. In conclusion, the resuits of this experiment indicate that the 17.02 Mev level in Be” has J = 1, even parity, and is formed from Lil by p-wave protons. The 18,14 Mev state has J = 1, odd parity, and is formed by s-wave protons. Ye THE STOPPING GROSS SECTION FOR PROTONS IN LITHIUM It can be seen from the preceding sections that, with the exnerimental methods outlined, the accuracy with which a scattering eross section ean he measirerd is no hetter than the aecursey with which the stopping cress section for the seattered particle in the target material is known. The stopping cross section for protons in lithium metal was given in 1938 by Haworth and King (16) for proton energies from 35 - 400 kev. They compared their results with the theoretical expression of Livingston and pethe 17) | and found the value I = 32 ev for the average ionization potential of lithium metal, Their experiment was performed by comparing thick and thin target yields, making use of the ratio of the stopping cross section of lithium to that of air given by nano ’!®) | Their result differed somewhat from the value I = 38 ev given by Mano for alpha particles in lithium, Since no measurements had been made in the energy range of interest in these experiments, it was decided to measure the relative stopping cross section for protons in lithium over this range. By normalizing the results to the theory at higher energies, the correct shape of the curve over the range of interest could then be determined. It was also hoped that the auality of the fit to the theory over the entire energy range measured would give an indication of the accuracy of the assumed value for I in the theory. ~ 38 = 1. Experimental Procedure, The method used was extremely simple. The equipment was first calibrated at several bombarding energies by measuring the energy of protons scattered from the surface of a clean copper target. A very thin layer of lithium was then deposited on the conver, end the measurements repeated, The proton energy lost in passing through the lithium layer before and after being scattered by the copper could therefore be determined. This work was performed at a scattering angle of 130 degrees. The scattering from clean copper was done at ion energies of 456 and 1000 kev, using the H and HHH beams, This experiment checked the previous calibration of the apparatus and indicated that the calibration was linear within 0.1 per cent, allowing the scattered proton energies to be calculated confidently at other vombarding energies, The thin lithium layer was then deposited, and the energy loss measured for bombarding ion energies of 450, + + 6C0, 800, 1000 and 1350 kev, using the H and HHH beams. 2. Calculations and Results, Assume a beam of protons incident on the target with energy Bape If the target is copper alone, protons scattered fron the surface will have energy E,.. where Eon * a(Cu) Eype a(Cu) is given by equetion 3,1i2a, in which M is now the mass of a copper nucleus, If the target is lithium on copper, the protons lose — FO energy fe, while traveling inward through the lithium and energy 4E, while escaping from the layer after being seattered; so those Bos scattered at the copper surface enter the spectrometer with energy Boy) where Eo, a(Cu) (Ey, LE, ) Lee Prebtons seatbered by the thin libhium layer during transit will have energies much less than Eng since more energy is taken by the recoil of the light lithium nuclei than by the recoil of the heavier copper nuclei, Consequently these protons are not accepted by the spectrometer, The energy losses ch, and fE 5 in the layor may be expressed in terms of the stopping cross section of lithium, so that a x = E, Ne cos 2, e(B,) Aw wT x a cod a, “s Gos 6, e(E,) where x is the thickness of the lithium layer, and E, and E. are some intermediate energies in the layer of the incoming and outgoing beams, respectively. Thus, if the target is set so that 2, = 8., a the measured erergy loss due to the presenes of thelithium layer is e = = s gl + AE t SE,, Eo, 7 Ep of, + A (5.1) = = (E,) + e.)] * COs 8, fac G 1? EXE o ie where « is a(CGu). Expanding e(Z,) and c(i) about some energy Ey» we obtain x “ _ de | aad = Yer oS, “s cos & (a #4) (By) aE. fs 1 2 7 Ca Ig 4 EoP x 1 a | 5 - a & te oun 4 ron Ea + ~~) Laz, ~ 3) > see -~ i 4 + see x a, * By Setting E. = 7 to eliminate the first order term, we have of, + Ey 6B, =N fat de = 2 s cos 8, \ ) a + | 1 ar 5 £ 0. are = ue tra za dE | B= EB x fan’ + pnd eon on Bo To determine Ey and ES we set c(E) = C/E 7 Over a small energy variation. a(B) ey emains essentially constant so that taking alZ,) = ays and a(Z..) = ao, we may write E . BL 1B oop 8 cos 6, t i c{E, ) E._<(E E(B) 2 “1B ay 1 ; a, att ie. Bo! os 7 R= fr ~ (Eo - AE . MBs Be ey ig 7 Fp Sy) | “ya ao =a meee cecovamenm mama oe | semen + see E,* Sip [ 173 F a (a) eT iB Ve (563) and similarly for Bo we obtain 1 2 1 "i @ AEs Ce ° = 2 \ ve Ei. = E. Tte- to (a, - 1) — + oee (54) If we ignore the second-order terms in equations 5.3 and 5.4, and use equation 5.1, wo may now write ar Z T= 0 3 BO6 ony ci] an 20 EO = , = + — x at a+ 4 2(a +1) LE, ——= + a Oey ® nO y nw boy Tne second term of this exoression was found to be about « per cent of the first term; so the second term was evaluated by setting 27 * s cy " ‘ 4 ee and taking values for e(E 0) and c{E, p) from the theoretical stopping cross section curve. The effect of the second~order terms in equations 5.2, 5.23 and 5.4 was investigated i fy ~O ! by successive approximation and found to be entirely negligitie for the accuracy attainable in this experiment, The expression used in calculating the final results was therefore Bop 7 B26 B i i a wo Te te Bon * Bo, Bop Bog %' 7 2 & 4 + a ? a +1 2(a + 1) 7 +a + where 3B = Ny aig and was a4 constant of the target angle and s 8, _ e(E,.) of the particular Lithium leyer, and where q id cE ° (Ey, The constant B was evaluated by normalizing the experimental results to the theoretical expression of Livingston and Rethe at a single point, E = 951.7 kev, using the value I = 32 ev given by Haworth and King. The fit of the remaining experimental points to the theory is shown in Figure 11, It can be seen that the fit is very good over the energy range studied, indicating that the theoretical expression is valid over this range. The theorelLical curve wilh I = 32 ev was therefore used in calculating the La! (pp) cross sections, The relative values measured in this experiment, which should be accurate to 3 per cent, are given in Table 9 in terms of the value et 951.7 kev. After this work was completed, it was pointed out to the author that recently an absolute determination of I for lithium 1 has been made by Bakker and segre (19) » who give I = 34.0 ev within about 10 per cent. This is seen to be in agreement with the result of Haworth and King. Since a difference of 2 ev in I makes a about 2 per cent in the Lithium stopoing cross section, 3 this agreement in conjunction with the results of the present 7 experiment indicates that the uncertainty introduced into the TA’ (pn) eross sections by the use of the I = 32 ev theoretical stopping cross section curve is of the order of 4 ver cent, TABLE 9 The stopping cross section of protons in lithium metal at various energies, referred to the value at 951, 7 kev, fo band eur £ ak E( kev; (951 o7) 40h 03 79 558.9 1.43 76o.3 1.17 95167 7.00 and ND? sO e a he Oo 4@ 199) oN An APPENDIX Te oe a 2 om mame aware) CORRECTION FOR SCATTERING FROM Liv aT FORWacD ANGLES, Sinec natural lithiwa metsl wae used for the tarsets in ia & : : + 2 ‘ this experiment, Li~ was present in its natural abundance, i.e., 0.075, The high resolution of the spectrometer made it possible te rm separate the protons scattered by La’ from those scattered by Lae at scattering angles greater than 0 degrees, However, at more forward angles particles scattered by both Isotopes were counted and a correction to the observed cross section was necessary. Let us denote the two isotopes, of abundances A and A*, by unstarred and starred symbols, respectively. If the detector were able to differentiate between particies scattered by one or the other isotope we would have, from equation 3.11, at given values of the bombarding energy and spectrometer setting, 4 rol Ko ZeR # (fax, 0) =— e(E,) (a* + *) n#{Ex, @) 3 1 * 5 > ath N 20VE.», 1 » po N Ne . where x = A, and r = Ax, 5, and By are the reaction energies “$s $ corresponding to Ey and Boo (equation 3.7}. However, if the detector does not differentiate between the two sets of particles it counts N, = nh tn If we are interested in the cross section zs pe - L5 = of the unstarred isotope, we therefore take ny =N -n Substituting this in the above equations, we obtein for the unstarred cross section do N* (a +) €(8,) ag* Gf oe) == 2 -— ~ (2,8) d 1 d ft NN (a* + 9%) eE{EF) dst i where ag/an is the unstarred cross section as calculated fron the observed number of counts, Expressed as the ratio to Rutherford seattering, we have r > Nx (a +») E(B, ) o*(E*,8) No (a* +4*) c(E¥) (BE, 28) For the separation of the lithium isotopes, taking Lil as the unstarred isotope, H#/N = A*/A = €,081, If the scattering from Lie is assumed to be given by the Rutherford scattering law, the remaining facters in the correction term give a product which for this exneriment was always very nearly unity. Therefore if the spectrometer accepts protons scattered by both Li! and Lie, the ratio of the Li scattering cross seecticn te the Rutherford cFoss section is given by o 7 (E,, @) = a - 0.08 R R This correction was apnlied to the results at @ , = 50 and 70 degrees, Cl The Lae Rutherford scattering assumption was checked at -~ 16 = 400 kev and found to be true within about 20 per cent, The resuits of Bashkin and Richards on Li’(pp) indicate that it is also very nearly true at 1200 kev for scatvering at 166 degrees. 7 However the presence of levels in Pe at proton energies near 906 kev (oroad) and 1820 kev throws some doubt at other energies, Consequently the Li “(pp) results at 50 and 70 degrees may be several per cent in error due to this ceuse. AT BEAM FROM E.S GENERATOR / if oma | / \ | / \ | Hi <> TARGET / a ai / oe PP ~ [ft ——— ral \GHAMBER \ \ mS SEE \ ~\ 3 2 FIG. 18 10% NCH a \& VARIABLE ANGLE \ \ | SPECTROMETER \ \\ I (IGO° POSITION) | — \\ FIGURE 1A- LAYOUT OF APPARATUS we \ SN oe a a ‘ NN \ NN, \ ON N NN “SS, t nh Life} i } i Pe. no Ww L _ ta Sa ‘ io” Eas O ROUSITION FIGURE 1B- TARGET CHAMBER DETAIL 30V4d 310d WONS HLVd 4O S3HONI bo Z~ Zz t 9 8 QO} rA| I L I } { L si i i i , ee) ee a ao 2'0- / tO - 39°0 + g0 4 1040 4 1.000 - 0.960 - 0.940 4 9.50 - 51 = INCIDENT BEAM A st & > ‘one oe a. Gm. / wes ~ Ny . , ‘9 Xo NU - \ a ec a . ; . \ N ~ = / _ Se . \ NN L E,- a) / ~ . NY ; ~ / NN SN / NN NN GL AB > EFFECTIVE ar JS > TARGET oN vA THICKNESS | A | Se a TARGET ——— MATERIAL NN J FIGURE 4 FNERGY RELATIONS IN TARGET O/ Op - 52- Los + a ee _ a _ + ene — 14 mt ca neo LL Fom = 50 os . i . ; nee be 7 a aan FIGURE 5 | 440 460 480 p (KEV) ™~ ~ | | Li’(pp) 2.2 oo o/ OR f | p18 || \ Gcm=!30° i / 1.4 y, " sae -0.6 L1 8 th ik K-14 ff Veer \ Oe asennad ngs Ep»(KEV) | -0.6/-— { 360 | 380 | 400 | 420 | 440 | 460 | 480 FIGURE 6 o/ Op 1.4 vi \ -1.0- ee enn all = Lig | LL O weep = Seared \ . 06 Ep (KEV) | fae aerate ete “360 , 380 | 400 | 420 440 460 480 i FIGURE 7 = 55 = do qa: IN BARNS/STER. Ep IN KEV — EXPERIMENTAL —— RUTHERFORD FIGURE 8 - 5 - do ~ dn: IN BARNS/STER. Ep IN KEV — EXPERIMENTAL —— RUTHERFORD ~ gf f Cap aa wo on Owe OOo ne LG oo Ocm=!30° —_ a p oT 800 1000 1200 FIGURE 9 - 57 = d IN BARNS/STER. | Ep IN KEV my, \ 8cmz!60° — EXPERIMENTAL — RUTHERFORD 600 —- 800 FIGURE !0 - 58 = T ¥ r | t T T 00d! OOO! OOS | OOS OOP 002 (nay) o3 a @ se : . E ‘ABM 286 LY BANND GIN0S OL ~ > Q3ZITVWHON ‘LNSWIM3dx39 SIHL- — 0 y ° é ONIN ONY HLYOMVH- — e 2 ; ] } b ‘N9ZE=T “FHLAG GNV NOLSONIAIT-3ANND GIN0S ‘\ re Ww (tl Sunoig) WNIHLIT NI SNOLOYd YOS NOILDSS SSOYD ONIddOLS ~ 60 — Li (pp) -8 FT| _F Ca on MAX —R'MIN (6) (7) (10) (17) (12) -61 - REFERENCES Cockroft, J.D. and E.T.S, Walton, Proc. Roy, Soc. 137A, 229 (1932). Lauritsen, C.C. and H.R. Crane, Phys. Rev. 45, 63 (1934) 6 Hefstad, L.R. and M.A. Tuve, Phys. Rev. 47, 506 (1935). Creutz, E., Phys. Rev. 55, 819 (1939). Fowler, WeA. and C.C. Lauritsen, Phys. Rev. 76, 314 (1949). Hunt, S.E., Proc. Phys. Soc., A, 65, 982 (1952). Devons, S. and M.G.N. Hincy Phyo. Reve 74, 976 (1948) 3 Proc. Roy. Soc. 199A, 56, 73 (1949). Brown, Snyder, Fowler and Lauritsen, Phys. Rev. &2, 159 (1951). Cohen, E.R., Ph. D. Thesis, California Institute of Technology, (1949). Snyder, Rubin, Fowler and Lauritsen, Rev. Sci. Inst. 21, $52 (1950). Tollestrup, A.V. Phe D. 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