The Elastic Scattering of Protons from Li⁶ Nuclei Citation McCray, James Arthur (1962) The Elastic Scattering of Protons from Li⁶ Nuclei. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2E98-TW57. https://resolver.caltech.edu/CaltechTHESIS:08242011-160444604 Abstract The differential elastic scattering cross section for protons from Li^6 nuclei has been measured for energies from 0.45 Mev to 2.9 Mev at six different angles. A measurement was also made of the Li^6(p,α)He^3 reaction cross section in order to determine its absolute value. The scattering data is consistent with an s- and p-wave phase shift analysis with a p-wave 5/2^- state at about E_p(Lab) = 1.84 Mev, with resonant parameters consistent with the parameters previously assigned to the corresponding mirror level in Li^7, and a very broad s-wave 1/2^+ state near or above E_p(Lab) = 2.76 Mev. The data is also consistent with a p-wave 3/2^- state with a different behavior of the s-wave scattering at higher energies, if an appropriate channel spin mixture is chosen. A p-wave 1/2^- state is not consistent with the data. No evidence for the existence of a 3/2^+ state near E_p(Lab) = 1 Mev with an appreciable Г_p/Г has been found. The stopping cross section for protons on lithium follows the Bloch formula from 0.8 Mev to 2.8 Mev. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Barnes, Charles A. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1962 Record Number: CaltechTHESIS:08242011-160444604 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08242011-160444604 DOI: 10.7907/2E98-TW57 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6620 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Aug 2011 14:39 Last Modified: 21 Dec 2023 20:24 Thesis Files PDF - Final Version See Usage Policy. 24MB Repository Staff Only: item control page

THE ELASTIC SCATTERING OF PROTONS FROM Li 6 NUCLEI Thesis by James A. McCray In Partial Fulfillment of the Requirements for the Degree of Doctor of Philos ophy California Institute of Technology Pasadena, CalifornI a 1962 ABSTRACT The differential elastic scattering cross section for protons from Li 6 nuclei has been measured for energies from 0.45 Mev to 2.9 Mev at six different angles. the A measurement was also made of 6 3 Li (p, a)He reaction cross section in order to determine its absolute value. The scattering data is consistent with an s - wave phase shift analysis with a and p- p-wave 5/2- state at about E (Lab) = 1. 84 Mev, with resonant parameters consistent with the p parameters previously assigned to the corresponding mirror level in Li 7, and a very broad s-wave 1/2+ state near or above E (Lab) = p 2.76 Mev. The data is also consistent with a with a different behavior of the p-wave s-wave scattering at higher energies, if an appropriate channel spin mixture is chosen. state is not consistent with the data. of a 3/ 2+ state near E (Lab) p 3/2- state A p-wave 1/2- No evidence for the existence =1 Mev with an appreciable r p /r has been found. The stopping cross section for protons on lithium follows the Bloch formula from 0.8 Mev to 2.8 Mev. ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. C. A. Barnes for suggesting this work and for his continued encourageInent and guidance throughout the past years. The advice given by Dr. R. F. Christy on the theoretical analysis of the data was also greatly appreciated. The author would also like to thank the faculty and staff of the Kellogg Radiation Laboratory for providing such a stiInulating atInosphere in which to work. The cheerful assistance of Mrs. Barbara ZiInInerInan in perforIning part of the calculations and the patient understanding and help froIn the author's wife, Diane, were Inajor factors in the bringing of this work to COInpletion. TABLE OF CONTENTS PAGE PART I INTRODUCTION. . . . . 1 II DISCUSSION OF THE EXPERIMENT. 5 A. B. 5 5 5 13 15 15 17 C. D. E. F. G. III IV . . . . . Apparatus. . . . . . . . Thick Target Technique 1. Derivation of Equations 2. Following Procedure . Calibration of Apparatus . 1. Electrostatic Analyzers 2. Magnetic Analyzers. . 3. Magnetic Spectrometer Resolution-to-Solid Angle Ratios. . . . . . . . . . . 4. Determination of Scattering Angles. Target Preparation . . . . . Stopping Power Measurements. Corrections to Yields. Experimental Results 1. Excitation Curves. 2. Angular Distrib')tions. . . 3. The Li 6 (P, a.)He Reaction. 4. Probable Error. . . . . . 17 1S 19 21 24 27 27 28 28 30 THEORETICAL ANALYSIS OF THE DATA 32 A. B. C. 32 38 42 Discus sion of the Theoretical Equation. Application to Li 6 (p. p)Li 6 . . . . . . Technique of Analysis of Data and Results CONCLUSIONS 53 APPENDIX I - Calculation of Measured Elastic Scattering Cross Section . . . . . 55 APPENDIX II - Determination of S- Wave Scattering Amplitudes . . . . . . . . . . . . . . 59 APPENDIX III - Considerations of the Possible 1/2- and 3/2- Assignments for the 1. 85 Mev p- Wave Resonance 62a REFERENCES 63 TABLES 65 FIGURES 74 -1- I. INTRODUCTION One of the most informative techniques available to the experimental nuclear physicist is the elastic scattering of charged particles from nuclei. The coherence of the various scattering pro- cesses gives rise to interference terms in the scattering cross section between the Rutherford amplitudes and the nuclear amplitudes. These additional terms frequently assist in the determination of many of the parameters describing the nuclear states involved. These parameters may then be compared to the predictions of nuclear theory. Perhaps the most interesting nuclei to study are the light mirror nuclei at low energies because here one usually has wellseparated anomalies in the elastic scattering and reaction cross sections which may be related to resonant behavior in the compound nucleus. The charge symmetry of nuclear forces may then be checked by comparing the level structure of the mirror nuclei. A very interesting case, which has already been the object of many studies, is that of Li7 and Be Selove, 1959). 7 (Brown, 1951; Ajzenberg- The first three levels of Be 7 have been well estab- lished experimentally, but the region between 6 and 8 Mev has not been fully studied. sample of Li 6 This region may be reached by bombarding a nuclei with a well-defined beam of protons and ob- serving the particles which corne out. be considered: The following possibilities must -2Li 6 He Be Li 6 + P -- Be 7* -- 3 7 + p (a) +a (b) + 'Y (c) Li 6 * + pi Be Li 6 5 He 4 (d) +n (e) +d (f) + d +p (g) The region of interest above corresponds to protons of energy between about 0.4 and 3.0 Mev. Although the three-body break-up threshold (g) occurs at 1.716 Mev, the energy in the center of mass system available to the three particles will not be sufficient over the region studied to allow the particles to penetrate the coulom.b barrier. the deuteron and the proton combine to form a He there is enough energy available for separation. of the three-body break-up com.pared to the a-He neglected over the region studied. 3 However, if particle, then Thus the contribution 3 break-up may be The (.e) aha (f) m.ode s need not be considered since there is not enough energy available in the compound nuclear system to allow such decays. The inelastic scattering process (d) . requires at least a proton energy of 7/6 (2.l84) Mev to reach the first excited state in Li 6 and at least d-wave protons, so this process would not be important except perhaps in the high energy part of the region considered. The radiative capture of protons (c) has. been ·studied (Bashkin, 1955; Warren, 195 6) up to 0.75 Mev and has been found to have a cross section of about 10 -31 2 cm.. Gam.rna rays were observed 7 to the ground state and to the first excited state of Be , and the -3- angular distributions obtained indicated that, if the process occurred through a compound state, it would have to have negative parity. McCray and Smith (196Z) have also l ooked at the radiative capture of protons by Li 6 at higher energies (1 Mev to Z Mev) and have found large cosZe terms in the angular distributions. However, the yield curve increases with energy without exhibiting resonant behavior. The reaction Li 6(p, a)He 3 (b) has been studied by several in- vestigators. (Bashkin, 1951; Marion, 1956) and all have found a broad resonance at about 1. 85 Mev and evidence of a broad structure at lower energies. There is, however, some confusion as to the value of the cross section, although all investigators obtain values in the region 10 -Z7 to 10 -Z5 Z cm. Marion et al. (Marion, 1956) measured angular distributions for this reaction and found large cos e terms which might indicate the presence of two compound nuclear states of opposite parity. They assumed the 1.85 Mev resonance to be formed by p-wave protons since they did not nave to consider terms higher than cosZe in their angular distribution. From the integrated cross section they were able to fit the 1. 85 Mev resonance with a single level Breit- Wigner curve assuming JW = 5/Z- for the state. With this assignment for the 1.85 Mev resonance, the other interfering state would t hen have + parity; and if formed by s-wave protons, would have to be 3/Z+, since the 5/Z3/Z spin channel. state can only be formed in the The elastic scattering of protons from Li 6 (a) was investigated by Bashkin and Richards (1951) only at one back angle with a natural Li target. They found an elastic scattering anomaly with -4a cross section of about 10- 25 2 cm • In the above region of interest it is seen that the radiative decay Il'lode is negligible compared to particle decay, so we have here a systeIl'l with essentially two channels, scattering and one reaction, and perhaps two broad interfering levels. It was the purpose of this investigation to Il'leasure experiIl'lentally the elastic scattering of protons from Li E p 6 nuclei in the range = 0.45 Mev to 2.9 Mev, to check the Li 6 (p, a)He 3 reaction, and to then analyze the above data in order to deduce the quantuIl'l nUIl'lbers of the states involved. -5II. A. DISCUSSION OF THE EXPERIMENT APPARA TUS Tliis experiment was performed in two parts. The low energy work from 0.45 Mev to 1.2 Mev involved the use of the 2-Mev Van de Graaff accelerator, 80 0 electrostatic analyzer, 2. 5-inch scattering chamber, 10. 5-inch· magnetic spectrometer and cesium iodide scintillation counter with amplifier and scaler, all of which have been previously described. 1950; Mozer, 1956) (Lauritsen, 1941; Fowler, 1947; Snyder, The work from 1.0 Mev to 2.9 Mev was accom- plished with the use of the 3-Mev Van de Graaff accelerator, 90 0 electrostatic analyzer, 6-inch scattering chamber, 16-inch magnetic spectrometer and a cesium iodide scintillation counter like the one above. The 6-inch scattering chamber with target holder and furnace was constructed during the course of this experiment and will be discussed in more detail in the section on target preparation. B. THICK TARGET TECHNIQUE 1. Derivation of Equations Consider now the situation described by figure lao A beam of particles of known energy EIB impinges upon a thick target. Some of the particles scatter from the front surface or produce reactions at the front surface., and the resultant particles leave with energy E 2B Others penetrate the target and scatter or interact at various depths inside. For any scattering or reaction event, relativistically by conservation laws to be E2 is determined non- • -6(1) where a l/Z = . . _+(w + .... Z)l/Z (Z) (M M )l/Z .. = 1 Z (3 ) (4) MO = mass of target nucleus ~ = mass of incoming particle M =mass of outgoing particle Z M3 = mass of residual nucleus EO = kinetic energy of target nucleus =kinetic energy of incoming particle E = kinetic energy of outgoing particle Z El E3 = kinetic energy of residual nucleus 9 =laboratory angle between incoming particle Ml and outgoing particle M Z The magnetic spectrometer will accept particles of energy E ZO 2: (~EZO/Z). The energy E ZO may be set so that C.E ZO corre- sponds only to particles emerging from a definite lamina C.S at a depth s inside the target. The spectrometer accepts particles which leave the target within a certain solid angle 0L. The number N of detected particies from a reaction or scattering process is proportiona 1 to the number of incident particles impinging on the target, -7 - which is equal to the total charge collected divided by the charge p er p article, to the number of target nuclei p er unit area facing the beam., and to the solid angle acce p ted by the m.agnetic s p ectrom.eter. The prop ortionality factor is defined as the cross section and is a function of the incident energy and the angle of observation. N where n Thus SS = cv r d«T(d,9) n ds dil cos 9 Ze J 9 cp s d 1 is the number of target nuclei p er unit volume, are p olar angles with respect to the beam. direction and Z and · 9 1 (5) 9 and cp axis, is shown in figure lao The stopping p ower dE) dE) is defined as 1 dE = - tr <IX (6) s where N s is the number of stopping atom.s per unit volume. From. figure la, one has then the following relations: 1 N rEI s 1 Ns where 9 2 JE dE dE) IB s = - -c~o-s""a"'l E20 dE E Em")-2 S s (7) (8) is defined in figure lao In order to proceed further with equation 5, one m.ust change variables from s to E 20 , giving -8- N = 'r \ SeJ~E CV II.. Ze cos 9 1 dCT(E1,O) (9) an 20 Equations 7 and 8 may be combined to give SE E s =- cos 0z N s S E 20 dE _ cONs °1 dE) - s 2 1 EIB dE dE) which may then be differentiated with respect to E 20 (10) to give (11) This assumes EIB and cos 01 constant. Now (IZ) Hence 1 (13) and (14) Thus N = CV Ze n Ns SoJ<Pr S E (15) ZO -9dcr ( E , e) is a slowly varying function of If <in l dEZO) is a slowly varying function of E 20 e and E , if l , and if 8E / 8E is a Z l slowly varying function of E , then equation 15 IIlay be integrated to l give N -- CV Ze where n (16) N" s dcr <in (E , e) now is the cross section averaged over the IIlagnetic l spectroIIleter window, and 0L is the laboratory solid angle seen by the IIlagnetic spectroIIleter. The energy spread ~EZO accepted by the magnetic spectroIIleter is then related to the mOIIlentUIIl resolution of the spectroIIleter R = lp by (17) The equation then for the experiIIlentally deterIIlined laboratory cross section is A further approxiIIlation IIlay be IIlade, if one looks at a laIIlina close to the front of the thick target. dEZ) :::: d E ZO Then dEl)::: dElB) and ) so equation 18 becoIIle s (19) For elastic scattering 8E /8E Z l = a which now is a function of angle -10only. One should see the thesis of Bardin (1961) for a more general discussion of the thick target equation. Since the above is the laboratory cross section at energy E , l one must find a relation between El and the experimentally determined quantities EIB and E E 1 dE S €\E) = E IB ZO . cos 9 Z cos 9 1 From equation 10 one may write SE ZO dE Z Now if dE) is a slowly varying function of E to El and E Z to E ZO (ZO) €"{'E) E over the range EIB ' then the following approximation is sufficiently accurate (Z1) where El and E Z are intermediate average energies. solve this equation for E Z as a function of E . l In order to E , one must then use equation 1 which gives l For the particular case of elastic scattering, equations 1 and Zl may be combined to give cos 91 cos 9 dEZ)E IB + d E 1)E ZO Z E 1 -cos 91 cos 9 d'E Z) + ad'El ) Z (ZZ) The stopping powers in this equation may first be evaluated at _ EIB +E l and E and an El determined. Then with dEl) E( 2 ) ZO _ EZO+aE l . and dEZ) d 2 ) a new El IS determined. This iterative = = procedure is continued until El is stable. If one is looking near the -11front edge of the target, usually the first approxiInation is sufficient, and as one moves back into the target, one or more iterations become necessary. For the case of a reaction one must first estimate what El might be, determine a(E ,6) from equations 2, 3, and 4, use l equation 22 to compute El and then iterate until stability in El is reached. Also for the case of a (23) in equations 18 and 19. In order to compare the measured cross sections with the theoretical equations, a conversion must be made to the center of mas s system, using (24) where (25) with y:: X sin 6 Z :: X L cos 6 L and x 2 ~J In the above work EIB and E 20 have been referred to as -12- Il1easured quantities. The energy EIB is that of the incoIl1ing beaIl1 of particles and is deterIl1ined by the use of a cylindrical electrostatic analyzer. For such an instrUIllent the energy of a particle passing through a circular equilibriuIl1 orbit is ~l + where r l analyzer, and r 2 (27) are the inner and outer radii of the cylindrical V is the voltage across the plates, incident particle, and M c O 2 Ze is the charge per is the rest energy of the incident particle. A sIl1all fraction of the voltage V is Il1easured by a potentioIl1eter, so equation 27 Il1ay be written (28) The constant C EA is then deterIl1ined by the use of a well-known resonant or threshold reaction. The energy E 20 is that of the outgoing particles and is Il1easured by the use of a double-focusing Il1agnetic spectroIl1eter. The energy of a particle passing through an instruIl1ent of this kind is E where B orbit. :::: (ZeBr)2 [1 _ I ZeBr )2 ] 2 20 2Moc '2Mo.c2 (29) = Il1agnetic field at particle orbit, and r = radius of circular -13The magnetic field is determined by a constant torque fluxmeter so equation 29 may be written (30) where The constant eMS is I is the fluxmeter coil current. determined by scattering particles of known energy from the front surface of a thick target of heavy nuclei. If at some fixed angle one plots the number of particles counted, for a definite charge collected, as a function of fluxmeter current, the resulting graph is called a spectrometer profile. Figures 2 through 6 illustrate such profiles. 2. Following Procedure The object of the experiment is to obtain the scattering cross section as a function of energy (excitation curve) and of angle (angular distribution). However, to take a profile at each energy and angle would be almost prohibitive in the time required. Fortunately it is not necessary, for one can pick a depth within the target, corresponding to some point upon the Li 6 profile and then follow along at approxi- mately the same depth within the target as the incident energy or spectrometer angle is varied. As sume that the difference between the energy of the particles seen by the magnetic spectrometer E and that of the particle originating in the front surface aE a certain fraction of the energy aE IB • Then IB 20 is always -14- aElB - E 20 = KaElB (31) or E 20 = (1 - K)aElB In terms of the fluxmeter current this becomes approximately (32) 1= [ Thus one decides on the value of K, usually 0.02 to 0.04, determines (C~S/(l-K) )1/2, and uses equation 32 to find the new fluJmeter setting as one changes either or both angle and energy. The choice of K is dependent upon the particular target. One necessary condition in the use of equations 18 and 19 is that the lamina contain only the element of interest or its isotopes, in this case Li 7 and Li • Consequently, if one of the contaminants C 12 or 0 16 6 has diffused into the target, one must be certain to follow far enough back so that the Li 6 is not diluted. The condition of the front edge of the target may be estimated by looking at the Li energy sides of the contaminants. 6 profile and the low A rounding off of the Li 6 edge and a low energy tail on the contaminants indicates diffusion (see figure 4). The amount of contamination is a function of the residual gas pressure in the target chamber, time, and the amount of charge accumulated at a particular spot on the target. -15- C. CALIBRA TION OF APP ARA TUS 1. Electrostatic Analyzers The 80 0 electrostatic analyzer was calibrated at the beginning and end of the first part of the experirrlent and the agreement in the values of C was found to be about 1/1000. EA obtained was C EA = 1.000 + 0.1%. the resonant y's from resonance. The method used was to observe ~9(p, ay)016 (Marion, 1961) at the E p = 872.5 -+ 0.4 kev One of the curves showing the gamma ray yield, from a target of thickness setting is shown in figure 7. The average value s, as a function of potentiometer The center of this curve . El is equal to ER + ~, so by combining this with the equation (33) where ~ El is the energy loss due to surface contamination and V T is the target potential with respect to ground, and equation 28, one obtains ER ER b V - ---orZ ) V+ r- ) 2M c EA R ZV:: (34) O where b = 'Zs + ~ El + Z e V T The first target was thickness s (35) CaF 2 evaporated on a copper backing and the was estirrlated by weighing the CaF 2' which was to be placed in a tantalum boat, and assuming the mass to be distributed evenly on a hemispherical shell at a known distance. The target was -16- estimated in this way to be about 2 kev thick to 0. 9-Mev protons. The target was m.ade in a bell j ar and then im.m.ediately placed in the scattering cham.ber so 6.El due to contaminant surface layers was assum.ed to be negligible. kept at +300 volts. For electron suppression the target was The second target was m.ade by holding a piece of copper over an HF bottle. The 90 0 electrostatic analyzer was calibrated at various tim.es 7 7 during the high ene rgy part of the experiInent, using the Li (p, n)Be threshold. * Either thin evaporated natural Li or LiF t argets on thick backings were used and the neutrons counted by a tillationi'couni.er. _ . a B IO plastic sc;in- An integral bias technique was used so that from. (yield)2 / 3 versus potentiom.eter voltage plot, shown in figure 8, the constant C EA could be obtained by extrapolating to zero yield. The necessary relation is Thus at threshold (37) VEA(l The value for Eth used was that given by Marion (1961) ; 1880.7 + 0.4 kev. = The average value of nine determ.inations m.ade at different tim.es during the course of the experim.ent was 2.274 Eth C 'E:A = 2:0.1% Me v/volt. *The author is indebted to R. K. Bardin for the use of his calibration data. -17 2. Magnetic Analyzers Both the 10. 5-inch and the 16-inch lTlagnetic spectrolTleters were calibrated by scattering protons frolTl clean copper surfaces. The calibration constant CMS will thus depend upon the electrostatic analyzer constant C EA and can be deterlTlined frolTl the equations (38) €2 - (a+ -)6.E + (l-a)ZeV €1 1 T (39) where the flwaneter setting I now is the value corresponding to half way up the profile. 2: 0.3 % ; therefore The variations in eMS were of the order the 16-inch spectrolTleter was calibrated on every run by scattering protons frolTl copper and that particular CMS was Jlsed for that data. The 1O .. 5-inch spectrolTleter was not calibrated with the use of copper profiles for every run; however, it was possible to calibrate the spectrolTleter f rOlTl t h e e 12 and 0 16 contalTlIDatlon pea k s and , 1Of 0 the contalTlination appeared to be slTlall, the Li used. 0 6 edges were also The lTlagnetic spectrolTleter constant used then was the average of the above values for each particular run. 3. Magnetic SpectrolTleter Resolution-to-Solid Angle Ratios A deterlTlination of the factor before equations 18 or 19 can be used. ZeR/2CVn L lTlust be lTlade This was accolTlplished by using - 18- for N in equation 19 the number of protons scattered by a clean copper surface, corresponding to a definite charge CV c ollected, and by assuming that the scattering cross section follows the Rutherford formula. The values of the Li 6 cross sections determined in this work, therefore, depend upon this assumption. For the high energy work this determination was made two or three times during each run and the average value used for that particular data. No special attempt was made to keep the trigger voltage of the integrator constant from run to run; however, the same capacitor was used for the copper scattering as for the Li 6 runs. For the low energy work different capacitors were used for the copper scatteri ng and Li 6 runs and the integrator ·firing voltage was measured once during each run. The low energy work, therefore, depends upon the measured capacitor ratios. 4. Determination of Scattering Angles For any particular reaction or scatteri ng cross section measurement, one first selects the desired angles in the center of mass system and then converts to the laboratory system by usin g the equation x + cos SCM sin SCM where x (40) is given by equation 26. For each scattering chamber the horizontal magnet angle SH was calibrated by first scanning the incident proton beam with a small aperture at a fixed distance from the center of the chamber to deter- -19mine a yield distribution as a function of angle"* Then the magnet aperture was scanned to determine the number of protons scattered, as a ·function of angle, into it. The difference in the centroids of the two distributions then gives 1T - 9 • H angle and at an angle near 90 be drawn. 0 This was done at a backward in order that a correction curve might The scattering plane for the 6-inch chamber was horizontal, but the geometry of the Z. 5-inch chamber, shown in figure 9, was more complicated. 10 o The particle beam enters the scattering chamber . 0 above the horizontal plane and leaves 10 below it. The equation relating the horizontal magnet angle (lH to the scattering angle 9 where here a:: 10 0 • A correction to 9 H was found necessary for both scattering chambers and was of the order of + 0.5 D. is (41) cos a sin sin L 0 to +10. TARGET PREPARATION Since Li forms LiZO and LiOH very rapidly in air, it is necessary to perform the actual evaporation somewhere in the scattering chamber. The targets must be smooth and they must not deteriorate too quickly under particle bombardment. Freshly evaporated copper on a clean microscope slide was found to be a very satisfactory backing material for the Li targets. The microscope slides provided the smoothness and the copper provided enough thermal conductivity to *The author is indebted to R. K. Bardin for the angle calibration data on the 6-inch scattering chamber. -20- prevent deterioration by local heating. The slides were cleaned first with a detergent and distilled water, then placed in chromic acid for a few minutes, rinsed in distilled water, and dried with lint-free gauze. The copper evaporation was performed in a vacuum bell jar and the copper blanks then placed i:m:mediately in the scattering cha:mber. A furnace was situated below each scattering chamber so that with the use of a long target rod, the Li evaporations could be carried out and the targets :moved directly up into the scattering chamber. Two. Lib metal samples were used, one of 94.5% purity and the other of 99.7% purity. purity sample. The Most of the work was done with the higher Lib metal was cleaned under kerosene, then transferred directly to the furnace and the system placed under vacuu:m. For the low energy work with the 2. 5-inch scattering cha:m- ber, a long narrow cold trap was used. tantalum sheets. The furnaces were made from However, satisfactory results were not always achieved with this set-up, so for the high energy work, improve:ments were made. Figure 10 shows a cross section of the b-inch variable angle scattering cha:mb·e r. In this set-up the second copper blank may be lowered through the cylindrical cold trap into the furnace area while the upper calibration copper blank is isolated from the Li furnace area by shields. The cylindrical cold trap was designed so that it was as near as possible to the target position for scattering. The better targets obtained with this set-up were possible :mainly because of the improved local vacuum. An additional feature of the new set-up was the use of a very pure carbon rod "cannon" type -21furnace which was heated by a coil of molybdenum wire. This furnace held the Li well, was very directional, and could be used many times. An indication of the quality of the targets obtained with the two different set-ups is found by comparing the contamination peaks in the profiles of figures 3 and 5. In the earlier work the Li gets were used for one or two days. finally developed so that the Li 6 6 tar- However, a procedure was targets were not used for a period longer than eight hours. E. STOPPING POWER MEASUREMENTS In order to use the thick target equation 18 or 19, one must first know the stopping power € as a function of energy. The values used here were those given in the review article by Whaling (1958). There was, however, some doubt about the values of the Li stopping powers for higher energies. Values deduced from old a-particle data of Rosenblum (1928) were considerably higher than that predicted by the Bloch equation. Consequently, a determination of the relative stopping powers of Li was made over the region 0.8 Mev to 2.9 Mev. The technique used was to scatter protons first from a clean copper target, then from a copper target on which a thin layer of lithium had been evaporated. Figures Ib and lc indicate the situation from which one obtains experimentally the energy displacement oE the copper edge. 20 of Figure 11 illustrates the displacement for two different spots on the same copper + lithium target. For a given tar- get spot the relative displacements are found for several energies -22- and relative stopping powers deterIllined. These values were then norIllalized to the low energy IlleasureIllents which are believed to be accurate on an absolute scale. Referring to figures Ib and Ie, one finds (Warters, 1953) (42) and (43) where Thus (44) FroIll the definition of stopping power Therefore (46) where (47) If now dEl) and €(E2) are expanded in Taylor series about SOIlle energy E, x one Illay write . N s - oE 20 = cos sa1 [(a.+I3)dEx ) + {a.(EI-Ex )+13{E2- E x )}~~ IE=E + " . x (48) -23where ~l-Ex and ~2-Ex are asswned small. Setting the coefficient a€ of the tiE term equal to zero gives (49) Now El and ~2 may be written as (50) and (51) So E x becomes I3~E2- aaE + l 2(a+l3) With the use of equation 44 and recalling E2B written as (52) = aElB , this may be ~E2 :: E2B + I3E 20 + 6E 20 [ E x a + 13 2(a+l3> 13(3'E.) - a AE~ ] (XE:"" ) + a (53) 1 Equation 48 becomes (54) The above two equations then are used to compute the relative stopping powers. In order to estimate the correction term in equation 53, one may use ~E2 €(E 20 ) "AEi : : . 13 (ElB ) € (55) -24and the theoretical Bloch equation (see Whaling. 1958) = dE) 2 aZ l Z 2 E E [In Z (56) + b] 2 For two different energies and the same target spot and geometry, equation 54 gives , dEx ) = dE ) (57) x The results of this determination are given in figure 12 and show that experimentally the stopping power for Li does follow the Bloch curve at higher energy. This curve was then used in the reduction of the 6 6 Li (p. p)Li . data. F. CORRECTIONS TO YIELDS There are three corrections which nlUst be made in this experi- ment in order to obtain the yield N to be used in equations 18 or 19. The first of these is a background subtraction. For this experiment the major background consisted either of protons scattered from the Li 7 contamination in the target or of a 3 reaction Li 6 (p, a)He • ++, s and He 3++' The Li7 thick target profile, shown in figure 4, was only present when the lower purity Li used. s from the 6 sample was The nature of the background at more forward angles may be determined from the profile in figure 6. is the unrestricted profile. The curve through the dots The crosses indicate the profile obtained when an 0.5 rnil Al foil is placed at the end of the magnet. The a 3 ++, and He s are stopped while the protons are energetic enough to ++J s -25 pass through the foil. The circles show the profile obtained 24 hours later. A second correction to be made is the loss due to high counting rate. Assume that th~ scintillation counter, consisting of Csl crystal, photo-multiplier, pre-amplifier, aIllplifier, and scalar, is dead for a time p seconds after each recorded event. rate is n R If the observed counting , then the fraction of a second during which the apparatus is dead is n RP • The fraction of tiIlle during vmich the apparatus is sensitive is 1 - nRP. events N • R This is then the fraction of the true nUIllber of Thus n R ::: 1 - n P R NR ( 58) or n N R- R (59) 1 - n RP The dead time p of the scintillation counter was obtained by scattering If, for a definite charge CV collected, one protons froIll copper. increases the counting rate n decreases. R , the tiIlle t required for the collection The difference tl. . in the nUIllber of counts recorded n. 1 1 and the true nUIllber of counts N is tl.. ::: N - n . ::: 1 1 or 2 n. p tl. i -- 1 n.p t.(l __1_ ) 1 t. 1 (60) -26An estiInate of the true number of counts is made by counting at a very low counting rate. the time t . recorded. The differences 1 recorded n. 1 lit.. 1 The counting rate is then increased and 1 in the number of counts and the above estimate for N is then plotted against This is shown in figure 13. at low counting rates 2 a slope equal to niP. the dead titTle ~. ~ From equation 60 one can see that may be approximated by a straight line with With the use of the equation p was found to be about 10 f.Lsec. This correction was found necessary for the copper calibrations and the forward angle Li 6 yields. A third consideration is that of charge exchange. proto.ns, Some of the 3 a-particles or He -particles, which scatter or are produced within the target, will pick up electrons on the way out and thus will not be observed in the magnetic spectrometer as protons or a's or He 3' s. Allison (1958) has measured the probability of this happening as a function of energy and gives data for various solids. In the present experiment corrections for this effect were less than one per cent for 3 6 proton scattering but were slightly larger for the Li (p, a)He reaction. There are two other important factors to be considered which may affect the yield. As the experiment is being performed one must check the operation of the scalar and the current leakage of the integrator. The screening effect of the electrons on the scattering cross -27 section was estimated by the Born approximation (Bohrn, 1951) for the Li 6 + P scattering and by a classical approximation (Wenzel" 1952) for the Cu + p both cases. G. scattering and found to be less than 0.3 % in Therefore a correction was not made for this effect. EXPERIMENTAL RESULTS 1. Excitation Curves The cross section for the elastic scattering of protons from Li 6 nuclei was measured from about 0.4 5 Mev to 2.9 Mev at intervals of 12.5 or 20 kev for c. m. angles near the zeros of the first and second Legendre polynomials and near the farthest back angle obtainable (- 160 0 ). The results are shown in figure 14 and indicate that the er scattering ler Rutherford scattering is Rutherford near 0.4 Mev and that rises gradually up to about 1.1 Mev where it then exhibits resonant behavior up to about 3 Mev. backward angle to 90 0 The decrease in cross section from the suggests that the resonance might be formed by odd i-wave protons. The cross section measured at 90 0 will then be the most informative, since the interference terms will vanish; this is shown in figure 15. A comparison of the 160 that of Bashkin and Richards (1951) at 9 L - 164 0 0 data (9 L - 156 0 ) with indicates that their cross sections are about 2/3 of the values measured in the present experiment. No indication was found of a large anomaly in the vicinity of 1 Mev. -28- 2. Angular Distributions The scattering cross section was also measured at c. m. angles o 0 0 . near 140 , 110 , and 70 at intervals of 100 kev. show the 25 angular distributions obtained. Figures 16 through 40 The error bars indicate relative errors and are 3% for the backward angles and 4% for the more forward angles 90 o 0 and 70. The scattering cross · sections deter- mined are given in Table 1. 3. 3 6 The Li (p,a)He Reaction In order to analyze the scattering data it is necessary to know the reaction cross section. Since the reported values (Marion, 1956; Bashkin·, 1951; Ajzenberg-Selove, 1959) for this reaction varied by at least a factor of 3, a new determination was deemed necessary. The thick target method was used and the angle of the spectrometer set at 9 L o , 3' :: 95 45 to detect the He s corning off in the backward hemisphere which correspond to the a I s which Marion (1956) measured at 9 L :: 60 in the forward direction 0 • The thick target profile obtained for an incident proton energy of 2.3 Mev is shown in figure 41. Since pulse height analysis was not used, the resulting curve is the superposition of the a profile and the He 3 profile. The two profiles may be separated with the use of the thick target equation 18. The positions at which the halfway points of the front edges of the profiles should appear are indicated by the lower arrows. fluxmeter current The spread in AI in which the profiles should rise was estimated from the equation 1 AE 601= "Z (-Ec-20 - ) ~/2 20 (62) -Z9where .6.E ZO /E ZO was calculated from (Cohen, 1949) .6.E ( E Z Z ZO ZO (63) ) Here R = iF = Z(l + M) zfr (64) the momentUIIl resolution of the magnet, and Olay be obtained from either this equation and the measured values of central path radius r, exit slit width .6.r and magnification M, or from the measured resolution-to-solid angle ratios if the solid angles are known. A value for .6.e was obtained from the Oleasurements on scattering 1 aE ZO was determined from the kinematical equation angle and E lIT)" ZO (Brown, 1951) 1 E For e L (65) ZO = 95 0 45' the calculated spreads .6.1 were determined mainly by the aEZO/ae terOl and are indicated by the bars at the bottoOl of the graph. From figure 41 one can see that the yield of the top of the He 3 profile should contain only He 3++ particles; therefore, this yield was used to calculate the cross section for the reaction ~i6(p ; He3)He4 at e L = 95 0 45', which should then be the same as 6 3 the cross section for the reaction Li (p, a)He at e L = 60 0 • When -30this was done it was found that the cross section was about 50 % of the value stated by Marion (1956) but in agreement with the results' of Burcham and Freeman (1950). By assuming the relative cross sections of Marion ' et al. to be correct and normalizing to the above value , one can then use equations 18 and 22 to predict the yield of He 3 ++, s at some depth within the target. This was done and a line drawn 'through this point and the yield at the front of the He file . The the He 3 3 pro- a profile' shown in figu r e 41 was then found by subtr acting profile from the total profile. The cross section f or 3 6 Li. (p, a)He was then com.puted from the a profile yield and was found to be about 60 % of the value quoted by Marion et al . Estimates of the reaction cross section made at other angles and energies f rom the He 3 profiles also indicated differences of about the same arn'ount. Therefo~e, the Li 6 (p, a)He 3 reaction cross sections of Marion et al. were assumed to be correct on a relative scale, and were normalized to 55% of the values quoted. The integrated reaction cross section 3 6 for Li (p, .a)He is shown in figure 42. These norm.alized values were then used in the analysis of the scattering data. A later reexami- nation of the work of Marion et al. turned up a factor of 1/ 2 so that their measurements are now in agreement with those of Burcham and Freeman (1950) and the present determination. 4. Probable Error The uncertainties of the relative stopping cross sections were mostly a result of the uncertainty in locating the mid~points of the copper profiles. The relative energy losses in the lithium layers -31were found from the difference of two measured quantities. the Hence % uncertainty depends upon the Illagnitude of this difference. i . e •• on the thicknes s of the layer. The uncertainties were found to vary from 2 to 7 % and are shown on. the stopping cross section diagram. Since the measured curve for the stopping cross section agreed with the Bloch formula. it was assumed that the stopping cross sections used in the determination of the scattering cross sections could be determined relatively from the Bloch curve to within an uncertainty of 2%. The absolute uncertainty for the stopping cross sections of protons on lithium and protons on copper was assumed to be 3%. the value given by Bader, et al. (Bader. 1956) . The uncertainties in the measured Li 6 scattering cross sections were estimated to be about 3% relative error for all angles. solute error was estiIllated to be about 5%. The ab- The sources of these un- certainties and their respective contributions are shown in table 2. and were combined as independent errors to give the values stated above. These uncertainties Illay also be taken as reasonable estimates for the pr obable error in the ratio. fT s catterlng . /fT u th er f or d' since R the coherent errors make only a small contribution to the com.b.ined error. -32- III. A. THEORETICAL ANALYSIS OF THE DATA DISCUSSION OF THE THEORETICAL EQUATION From the basic concepts of non-relativistic quantum mechanics one can derive the cross section for the elastic scattering of charged particles with spin i a scattering matrix from charged nuclei with spin I in terms of SJ a, s '1 ' . ,as 1 which relates the amplitudes of the outgoing waves to those of the incoming waves. section is also found in the same manner. spin s The reaction cross The concept of channel is introduced such that - =- s I +i (66) s= II-il • •••• I+i A given set of two particles is denoted by the . index a and is referred to as a channel. - For relative orbital angular momentum 1 - pound nuclear state may have a total spin J a com- such that - =- +J s 1 (67) J=ls-1 1•• • •• s+1 For each possible state of given J one may associate amplitudes for various modes of formation and decay. These amplitudes are complex and may be related to a resonant description of the state. The object of the analysis is to deduce the various parameters of the resonant formulation. The technique of analysis of elastic scattering data used is that described by Christy (1956) and Mozer (1956) and involves the -33- display of the am.plitude m.atrix [A 0.0., s I S, V I]. v Here the index v designates the m.agnetic substates of the channel spin s and the prirn.es indicate the outgoing channel. An elem.ent of the am.plit ude m.atrix is given by A ·t 0.0. , s's , vlv x (1 =,[Re1",o . r. 0 SIS v'v +!::!!!... 2: ka (21+1)1 /2 i(Ti +rJ ,-2Ti ) ell 0 SOV/JM)(llslrn.l,v'/JM)f~s'l1,y~l' (68) where ?ss'll' -- SJas 'l ' ; asl - 0 SIS . 0 1'1 (69) The first term. is the am.plitude for Rutherford scattering with f _=n---..,--,~ - ' 2 k sm . 29 a '! 2 R - ) (Rutherford cross section) (70) s= (Rutherford phase shift) (71) n= where v a (72) =hla /IJ. is the relative velocity, JJ. the reduced rn.ass and a the center of m.ass angle. 1 Til - TiO = I tan -1 n .... J (Coulorn.b phase shift) (73) j=l The second term. is the nuclear part and consists principly of a Cle bsch Gordan coefficient for forrn.ation of the state (1 so v /JM) and (1 fs lrn. ,vl/JM), the spherical harm.onic 1 m.£1 J Y l' of the exit channel, and the nuclear arn.plitude fssll1 f for one for decay of the state form.ation and decay of a nuclear state of definite spin J. The sum. -34in: equation 68 is' (74) where one should keep in mind that for an incoming beam along the z axis m! = 0 so that M = v ::: m!,+ v'. The differential cross section for scattering from magnetic sub-state v of entrance channel spin s to magnetic sub-state v' of exit :channel spin s' is found by squaring the appropriate element of the amplitude matrix. dUas'v',asv::: IA aa ,s's,v'v I2 (75) The differential cross section for scattering from channel spin s to s r for an unpolarized beam, is found by averaging over the magnetic sub-states of s and summing over those of sr. s 1 dUas', as ::: 2s+l sI L I (7, 6) v=-s vf:::_s' If in addition the particle detectors are spin insensitive, one must average over initial channel spins and sum over final channel spins. I'+i I I+i dUa,a::: L (2I+i)(~\+1) dUasl,as ') s;YI-i l s'::: II'-i' l (77) The final equation for the elastic scattering cross section is then IIH' I+i 1 du a, a::: (2I+l)(2i+l) s Sl S:::~-i S~~Iql v~s V~-Sl I I (78) -35- For a particular incident particle and target nucleus the anlplitude nlatrix elenlents are worked out for the lowest orbital angular momentuzn waves which are thought to contribute to the scattering. 6 6 The aznplitude matrix for the case of Li (p, p)Li with only s- and p ': waves contributing is shown in figure 43. The scattering amplitudes within each element or box are coherent and give rise to cross product or interference terms when the matrix eleznent is squared. The final cross section is then found by squaring each box, adding them and dividing by (21 + 1)(2i + 1). The elements of the scattering matrix SJ '1' 1 as ; as are complex quantities and are related to a resonant description by the relation (see Mozer's thesis (Mozer, 1956) for a more cOnlplete discussion). (79) where 6 = cot -1 E R - E r/2 (resonant phase shift) (80) q, is the Dpotential phase shift- which for the case of a charged hard sphere would be q, = - tan -1 Fl (ka) G (ka) (81) 1 where a is the -interaction radius. - The latter quantity is not well defined but is sometiznes given by a l 3 =R0 (Al / 3 + A / ) 10 with RO =1.45 x 10 -13 cm (82 -36F 1 (ka) and G (ka) are the coulomb wave functions evaluated at the 1 interaction radius a and wave number k. q, will also contain the "tails In general, however, states at energies far from the region being considered. ll of other The relation 79 enables us to separate a rapidly varying energy-dependent behavior in the scattering from the slowly-varying non-resonant scattering. The resonant parameters of a state are the resonant energy ER the total width r rp the proton width other partial widths r, r y , r n , ... a The partial widths are a measure of the probability of the state decaying via a certain mode and thus the partial widths must add up to the total width for decay of the state. r=rp +ra+r +rn + ... y (83) For a given mode of decay the nuclear part of the probability for decay may be separated out by defining the nreduced" width. where i denotes a particular channel and 1 momentum wave involved. k. 1 denotes the angular is the relative wave vector for the chan- nel i. 211" k.:: 1 A.. 1 21Tp . 1 :: -n= J.1.v. 1 1 (85) -37where v . is the relative velocity of the two particles involved, and 1 the "reducedmass n for channell PI (86) is the penetration factor and depends upon the relative wave factor, the angular momentum involved, and the interaction radius. where FI is the regular Coulomb wave function, regular Coulomb wave function. and G 1 is the ir- For a very broad state the level shift of the state (Thomas, 1951) must be considered. The resonant energy is equal to the . actuaL ' energy of the state (in the CM system) plus a level shift in energy. (88) where the index >.. denotes the state. The level shift is given by 2 'iU d(ln AI) Do i1 = -a.- [1 + d(ln k.a . ) ) 1 The quantity Al 1 (89) 1 is just 2 2 1/2 A (k .a . ) = [F (k.a.) + G (k.a.» 1 11 1 11 1 11 (90) The total level shift then is the sum of those for each mode of decay. (91) The partial width of decay is given by -38- (92) From equations 79 and 81 one sees that since ¢ is negative for "potential" type scattering; the complex points describing the scattering matrix element begin at (1,0) and move in a clockwise direction on the unit circle as the relative energy of the incoming channel increases. From equations 79 and 80, however , one sees that the complex points describing the resonant scattering move in a counter-clockwise direCtion (6's are positive) and the distance of the points to the point (1,0) depends upon the value of rp/r. resonance, 6 becomes fT /2 When E = E R , i. e., at and thus for pure resonant scattering the complex points describing the resonant scattering should cross the real axis at 1 - 2(r p If r p In. / r is close to unity and the potential scattering small, then the resonant complex point should be near the point (-1,0). r p If / r is small then it will be in the vicinity of the point (1,0) and will depend mainly on the potential phase shift. B. 6 6 APPLICATION TO Li (p, p)Li For this case one must combine a proton of spin and parity 1/2 + to a Li 6 nucleus of spin and parity 1+. possible channel spins, 1/2 then nuclear states in Be parities of 7 + If one considers only s-waves, may be formed which have total spins and l' =1/2+ or 3/2+. waves is 1-. + and 3/2 • Thus there are two The spin and parity associated with p- Thus nuclear states may be formed through the 1/2 spin -39- = l/Z - and 3 /Z -. States which may be formed . through the 3/Z spin channel have i" = l/Z -, 3 /Z - and 5 /Z -. The channel which have J'1f above possibilities then must be considered in constructing the amplitude matrix shown in figure 43. The experimental data indicates the presence of a strong resonance in the vicinity of 1. 85 Mev. The angular distributions show a decrease in cross section from backward angles to 9 CM = 90 0 • If only s- and p-waves are assumed to be important in this energy region then this nuclear resonance would have to be formed by pwaves. * The spin and parity of this state then nlUst be either 1/ Z-, 3/Z- or 5/Z-. The assumption will be made that there is only one p-wave nuclear resonance in the region investigated. The mirror nuclei Li7 and Be 7 shown in figure 44 have cor- responding levels for the first and second excited states. There is also a well-defined state at 7.47 Mev in Li7 which should correspond to the resonance seen in Be 7 at 7.18 Mev. cross section for neutrons on Li 6 From the total absorption and the cross section for the re- 6 3 action _Li (n,a)H , the spin of the above state in Li7 was found to be 5/Z- (Johnson, 19 54). It was not possible because of the divergent cross section:at f orward angle.s · and the unknown s-wave background, to 6 6 6 3 perform a similar analysis in the case of Li (p, p)Li and Li (p, a)He . However, it seemed reasonable to assume that the state in Be 7.18 Mev is the mirror state of the 5/Z- 7 at state in L/; therefore, 6 6 an analysis was performed for the Li (p, p)Li scattering under the assumption that only s- and p-waves contribute and that the p-wave The reduced proton width calculated by assuming f-wave exceeds the *single particle limit. -40scattering is only through a 5/2- resonant state. The amplitude matrix for this case is shown in figure 45 and is of ·course much simpler than that for the more general situation of s- and p-waves. The differential cross section for scattering is found by squaring each box, adding them together and dividing by six. There are three un- known complex amplitudes which describe the scattering, one for the resonant p-wave state and t>wo for the s-wave scattering, one for each spin channel. The real and imaginary parts of these complex ampli- tudes are written as follows: I lI I -Z -Zss (93) 5 f; 3 'Z-ZPP = f5 + ig 5 - I -Z -Z It is also convenient to define the quantities X, Y and U in terms of the above amplitudes. X= (94) U =I - (95) -41- It is also to be noted that 2 k (T R (s-wave) U = (96) 1T where (TR is the integrated s-wave reaction cross section. The dif- ferential cross section for scattering may then be written in the following way: d ..[R 1 ..[R U (T (e, E) = R + (,-sin £ - -2)(X-l) - -k cos £ y - --..,.. er ~~ K 2k 4k G + g3[ (is -l)sin 2(11 1 -11 0 ) + gscos 2(111 -110 )]} cos e 'Z "Z 'Z - {R {(f -l)sin[ 2(11 -11 )-£] S 1 0 'Z 9 ' 7 2 r + 50 k 2 (1 + b cos e)[ (f (97) + gscos[ 2(~-110)-£] }cos e 'Z 1) 2 2 + gi ] The first line is the s-wave scattering in the form given by Christy (1956). The second line is the s-wave, p-wave interference terrn. The third line is the p-wave, Rutherford interference term, and the fourth line is the' resonant contribution to the 5/2- p-wave state. The object of the analysis is to find a set of six coefficients, f 1 / 2 , gl/2' f 3 / 2 , g3/2' f S / 2 ' gS/2' as functions of energy which describe the 25 angular distributions over the energy region considered. These coefficients :must, however, also be consistent with -42the reaction data for 3 6 Li (p. a)He and they must vary smoothly as a . function of energy. C. TECHNIQUE OF ANALYSIS OF DA TA AND RESULTS An extension of the method given by Christy (1956) was used to determine a set of the six coefficients at each energy. Equation 97 may be written in terms of X and Y as a straight line. The energy will now be considered fixed and only the angle e will be varied. y .: A(e)X + C(e) (98) where the slope is given by A(e) : sin S - 1 lk'lR (99) cos S and is only a function of kinematical variables and the atomic numbers Zl and ZOo Recall that sand R are functions of angle. The intercept C(e) may be written as the sum of two parts. c(e) : B(e) + D(e) (100) where cos is the s-wave intercept. S (101) This term involves the Ineasured scattering differential cross section dtT(e)/dU and the integrated s-wave reaction cross section tTR (s-wave). The intercept D(e) contains the contribution of the p-wave -43resonant scattering, p-wave, Rutherford interference scattering and s-wave, p-wave interference scattering. Since the 5/2- p- wave state is formed only through the 3/2 spin channel, the s -wave, p-wave interference term contains only the s-wave 3/2 spin channel amplitude. The intercept D(e) may then be written as follows: D(e) = ZG(e) + H(e) (102) where Z = (f3 - l)S + g3 '2 (103) '2 is not a function of angle but only a function of energy. S, G(e) and H(e) The quantities are given by (f S - 1) cos 2("1-110) - gssin 2("1-"0) ______________7~_______ S=~7 (104) (fS - l)sin 2("1-"0) + gscos 2("1- "0) 7 7 (105) +_....;.9__ (106) sOk/R In this experiment angular distributions were taken at six different angles, hence six straight line equations may be written in terms of X and Y. The complex scattering amplitudes are restricted by the -44conservation of particles to lie within the unit circle in the complex .plane. This then results in restrictions on X, Y, and U. These are -1 ~ X ~ 1 -1 <: Y ~ 1 (107) o ~ U~ 1 The slopes A(9) of the six lines are calculated and the intercepts B(9) are computed. s-wave If the scattering is describable in terms of s-wave protons alone, then the six lines must intersect within experimental error at some point in the complex plane, which is just the point (X, Y) for this ene rgy. do intersect as shown in figure 46. At low energies the six lines The s-wave intercepts B(9) were computed for each of the twenty-five energies and the six lines for X, Y plotted in the complex plane. At higher energies the six lines no longer intersect within experimental error, indicating that s-waves alone are not sufficient to describe the scattering. The plot of the six straight lines with s-wave intercepts at an energy near the peak of the assumed 5/2- p-wave resonance is shown in figure 47. The next step was to assume reasonable parameters for the p-wave amplitude f / + ig / and calculate p-wave intercepts D(9) 5 2 5 2 for the six angles. The resonant description given by equation 70 was assumed for the p-wave amplit.u de for a state with J1I' = 5/2-. The p-wave potential scattering was neglected in the analysis since no good estimate of its contribution could be made. The resonant phase -45- shift certainly dominates over most of the resonant region. With this . assumption the parameters f / and g5/2 become: 5 2 r -t . (cos 205- 1) -Z (108) with (109) A reasonable choice must be made for the resonant parameters r p' 6 3 From the Li (p, a)He inte- r, and ER as a function of ene rgy. grated reaction cross section (figure 42), one may find values for the reduced proton width '{2 and the reduced alpha width '(2. p a The p-wave reaction cross section is given by O"R(p-wave) =!z (110) k and becomes at resonance (lll) This combined with the relation r = r p +r a yields two sets of values for rp and r a for an assumed s-wave reaction background, one set with large r p and small r , the other a -46with small r p and large r.a The magnitude of the resonance in the r p is large. .scattering data indicates that These two modes are the dominant modes of decay for the state. The relative wave vectors for the p_Li 6 system and the a_He system are given in terms of the incident proton energy El in the laboratory system as 2 24 -2 kp = 3. 536E x 10 cm l (113 ) k~ = 7. 027[ El + 4. 692] x 10 24 cm- 2 The parameters n defined in equation 72 must also be known in order to compute the penetration factors. n 2 These are given by 0.2248 = p El ~-=-- (114) n 2 a = El 0.7937 + 4. 692 In all of the above expressions the laboratory proton energy El must be expressed in Mev. Since the angular momentum quantum number of the incoming proton with respect to the Li 6 = 1, the nucleus was as sumed to be 1. penetration factor for the entrance channel is given by equation 87 with 1 = 1 and k p as given above. the p_Li 4.08 x 10 6 The interaction radius a system was taken from equation 82 whi ch gives a -13 cm. p p f or = The penetration factors were found from the graphs given by Sharp, et al. (Sharp, 19 55). The relative orbital angular momentum of the a-He 3 system 3 -47which results from the breakup of the Be JiI' = 5/Z- 7 nucleus in a resonant state is found from the conservation of angular momentum and conservation of parity. 0+ and the He 3 Since the a-particle has spin and parity spin and parity l/Z+, the only possible exit channel spin quantum number and parity is s' = l/Z +. Thus from equation 67 the lowest possible relative orbital angular momentum quantum number with negative parity in the exit channel is 1.' = 3 -. The penetration factor is then given by equation 87 with 1.' = 3 and k a as given above. The interaction radius was taken from equation 8Z which gives a = 4.39 x 10- 13 cm. a The reduced proton width '{ Z Z and reduced a-width '{a were p found from equation 84, which are in this case r P = Zk p P l '{ pZ Since the magnitude of the and r a = Zk a P 3'{ Za (115) s-wave background is not known apriori, various choices may be made resulting in sets of reduced widths '{Z p Z and '{a. These values, however, are very close to the "best" values for the reduced widths '{Z and '{Z of the corresponding mirror state n a in the Li7 nucleus as given by Gabbard (1961). The discrepancy between the previously reported value of '{Z for the Be a (Marion, 1956) and the '{ 7 * state Z for the corresponding Ll.7* state (Gabbard, a 19 61) appears to be a result of a computational error. Since there was no criterion for making a different choice, Gabbard's values were then used to compute the resonant parameters of the . B e7. state 10 5/Z- p-wave The resonant energy of the state was determined from the elastic scattering data at 90 0 • This data is shown in figure 15. At -48this angle the p-wave interference terms of equation 97 disappear leaving only the resonant contribution of the p-wave state. (116) This term may be written as a function of the resonant phase shift as follows: r 18 = 25 k 2 ( -/) 2 . 26 5 (117) SIn "2 p r and the partial widths r p and r a weTe The total width calculated as functions of energy using Gabbard's values for '{ '{2 and equations 112 and 115. a 2 p and Since this state is very broad the reso- nant energy was computed from equations 88, 89, and 91 which become in this case (118) I:::. X. = I:::. P + I:::. a '( I:::. 2 d(1n AI) =-..:..E[l+ ] p a d(ln k a ) p '( I:::. a 2 p P (119) d(ln A ) 3 = ~ [3 + dUn k a )] a a a The above level shifts were calculated with the aid of the graphs of Coulomb wave functions of Sharp et al. (Sharp, 1955), and were 2 determined from the slopes of the Al versus ka graphs. The resonant energy Ex. ' was chosen so that when the calculated p-wave -49resonant cross section equation 117, is added to the calculated Rutherford cross section, equation 70, the resulting curve is symmetrically placed beneath the measured curve. The p-wave reaction cross section for the 5/2- state may also be written in terms of the resonant phase shift . 2 r- SIn Us (120) 'l This was calculated as a function of energy and subtracted from the measured integrated reaction cross section to give the s-wave reaction background ITR (s-wave) which is necessary in the computation of the scattering cross section (see figure 42). The p-wave scattering amplitudes were then calculated for the twenty-five energies from equation 108. The variation of this scattering amplitude in the complex plane ' is shown in figure 48. The coefficients H(e) and G(e) were then calculated for the six angles at each of the twenty-five energies. The coefficient D(e) cannot be calculated completely since the factor Z given by equation 103 contains the unknown s -wave 3/2 channel spin scattering amplitude; however, Z at a given energy must be the same for all angles. If a scattering cross section can be described in terms of swaves only, then at each energy all of the straight lines must intersect at some point (X, Y). If the reaction cross section. is also known, then the function U is determined. For the case of two channel spins there are two complex scattering amplitudes and hence four unknown s-wave parameters. The quantities X, Y, and U defined by equations -5094 and 95 give only three relations between the four unknowns. How- ever, if there is a p-wave state present, then the s-wave, p-wave interference term will give a fourth relation betwee.n the unknown swave parameters. In this particular case the relation is given by the coefficient Z as defined in equation 103 and must also be determined from the scattering cross section. Of the four equations re- lating U, X, Y, and Z to f l / Z' gl/Z' f 3 / Z' and g3/Z' three are linear and one quadratic. The unknowns f / and gl/Z m.ay be l Z elim.inated from the equations 94 and 95, giving an equation for f / Z' 3 g3/Z which is just a circle in the complex plane with the center at X, Y and the radius a function of X, Y, and U. (f3 - X) Z 1 Z Z + (g3 - Y) Z :: "2 [ (1- U) - (X + Y )] "2 (lZl) "2 The equation 103 is just a straight line for f 3/Z' g3/Z in the complex plane with a slope dependent only on the p-wave state and an intercept which is a function of Z . g3 :: -Sf3 + (S + Z) "2 (lZZ) "2 If a solution for f / ' g3/Z is possible, the straight line must inter3 Z sect the circle and will in general give two solutions for the 3/Z complex scattering amplitude. The restriction that the complex scattering amplitudes (fl / Z' gl/Z) and (f 3 / Z' g3/Z) must lie within the unit circle, the restrictions on (X, Y. ), and the requirement of smoothly varying curves for the points (fl / Z' gl/Z)' (f3 / Z' g3/Z) and (X, Y), in the -51com.plex plane as a function of energy provide a basis for m.aking a choice between the two solutions. The com.plex point (fl / Z' gi/z) m.ay also be found geom.etrically, from equations 94, by constructing a straight line from the point (f / Z' g3/Z) through the point (X, Y) to 3 a distance which is twice that of the distance between (f / ' g3/Z) and 3 Z (X, Y). The procedure used to determine the figures 47 and 49) was as follows: s-wave param.eters (see If t h e six straight lines for X, Y do not intersect at som.e point in the complex plane within experimental error, then the coefficient H(e) was added to each B(e). intersect, then this would indicate a value of zero for not, then a value of Z If they do may be chosen and the six coefficients ZG(e) These are then added to the intercepts B(e) + H(e). determined; the lines still do not intersect, a different value for and the procedure repeated. value of Z Z. If the lines Z If m.ay be chosen If the lines do then intersect for som.e and the point (X, Y) is within the circle (1 - U)l/Z, and the straight line for f 3 / Z' g3/Z intersects with the circle for f 3 / Z' g3/Z and the points (f3 / Z' g3/Z) and (fl / Z' gl/Z) lie within the unit circle, and all three of the complex points are a smooth extension of the three curves constructed previously at lower energies; then this solution is taken to be an acceptable one. The s-wave, p-wave interference term. in the cross section may be written as IT where IT sp (E) sp (E, e) = IT sp (E) cos e is related to the coefficient Z (1Z3 ) by -52- (124) The variation of this quantity as a function of energy was computed from the Z's determined in the analysis and is shown in figure 50. The above technique was then used, beginning at low energies and proceeding to higher energies, to determine an acceptable set of swave parameters ~/2' gl/2' f 3 / 2 and g3/2. The resulting complex points are shown in figure 48 along with the points (f / 2 , g5/2). 5 Table 3 (X, Y) and gives the numerical values of these quantities obtained in the analysis. At each energy these six values for f l / 2 , gl/2' f 3 / 2 , g3/2' f / , g5/2 were substituted into equation 97 for each of the six angles 5 2 and values of ~ / R obtained. The solid curves drawn on each angular distribution represent the results of these calculations. Given this set of s-wave parameters as a function of energy it was then possible to calculate the s-wave scattering cross section at 90 0 as a function of energy and to add this contribution to the sum of the Rutherford scattering cross section and the resonant p-wave scattering cross section to give the scattering cross sections predicted by equation 97 at this angle. The result is shown in figure 15 and ap- pears to be a reasonable fit to the measured scattering cross section. -53IV. CONCLUSIONS Over the region studied, equation 97 appears to describe the measured elastic scattering eros s section. The p-wave scattering is consistant with resonant scattering through a state of spin and parity J?r = 5/2- with resonant parameters in agreement with those from the mirror level in Li cross section.* The 7 and with the 6 3 Li (p, a)He reac tion s-wave scattering phase shift in"the 3/2+ spin channel (figure 51) does not show a resonance in the region from 0.45 to 2.9 Mev but qualitatively follows the energy variation of charged hard sphere scattering with an interaction radius of about 2 fermis. Hence. if a 3/2+ state exists at about 6. 35 Mev excitation in the compound nucleus Be 7. it must have a small large r a /r. r p /r and a This state should then show up in the elastic scattering 3 of He 's from He 4". It should also be noted that the mirror level of 6 7 this proposed state could not be excited in the Li (d. p)Li reaction. (Hamburger. 1960) This evidence also suggests a small value of r n /r for this state. Of possible interest here are the cluster model calculations of PearlBtehi' (,l96G) and K b.a \1na 119611) who J! onc1ude~ that neither an alpha-particle plus mass-three clustering, nor a neutron plus Li 6 cluster, will yield the proposed 3/2+ state. The p-wave 5/2- state at about 7.2 Mev excitation is now usually assigned the configuration 4 P5 / , (Inglis. 1953; Marion. 1957) on an L-S 2 coupling model. These calculations also predict a state with con- figuration 2F 5/2 somewher.e in the excitation range. -5 to -7 Mev. *The possibility- of a 3/2- assignment for the p-wave state cannot be excluded. Al/Z- -aasig ~mentJ.ts " inconsistent with the data.(See Appendix III. ) -54If such a state falls within the region corresponding to 0.45 Mev <::; E p <::; 1.4 Mev, the present experiment suggests that the state must have a small value of r /r. P (See figure 46. ) The s-wave scattering phase shift in the 1/2+ spin channel has qualitatively the same behavior as that in the 3/2+ channel at low energy but begins to exhibit resonant behavior at about 1. 5 Mev and goes through £>::: .,,/2 at about 2.76 Mev. This suggests the presence of a broad 1/2+ s-wave state in Be 7 at an excitation energy of about 8 Mev, although the resonance energy may not be accurately given by the present analysis. Resonant parameters for the three states discussed above are listed in table 4. These parameters are consistent with the measured elastic scattering cross section for reaction eros s section for Li 6(p, a)He 3. 6 6 Li (p, p)Li and the The parameter s for the 3/2+ state (proposed by Marion to explain the energy and angular variation of the Li 6(p, a)He 3 reaction) were obtained from the re3 6 adjusted Li (p, a)He integrated reaction cross section, using the scattering data to make a choice between the two possible sets of values for r p /r and r a /r. The 1/2+ state seems necessary to understand the scattering data in terms of s- and p-waves alone. An analysis of the scattering data of Harrison and Whitehead (Harrison, 1961) in terms of s-. P-. and d-waves should shed further light on this region. level in Li 7 The parameters for the p-wave 5/2- mirror as given by Gabbard (1961) are also shown in table 5 for comparison. -55APPENDIX I Calculation of Measured Elastic Scattering Cross Section An example of the reduction of the measured yield to scattering cross section will be given for the data shown on the Li 6 profile in figure 5. The relation between yield and laboratory cross section is given by equation 19 as follows: N E 20 where a = The resolution-to-solid angle ratio and the factor Ze/ 2CV was determined as a product (ZeR/2CVQL) by scattering protons from copper just before and after the particular Li 6 run. The energies ElB and E 20 for the copper scattering were determined from equation 28 and 30. ElB = 1. 298 Mev E 20 =1. 252 Mev = 81 o13; a = 0.9734; 9 = 82 o14 CM 9 Cu(p, p)Cu I L 13 =1 I dQ CM dti L =1. 0047 from equations 25 and 26 The Rutherford scattering cross section equation 70 may be written -56- The laboratory energy E1 was cOnlputed fronl equation 22 which beconles in this case where €(E ) 2 ,,=-- = E(E ) 1 Fronl the stopping cross section curve for protons on copper (Whaling. 1956) E(E €(E 1B 20 ) = 11.1 x 10 -15 eV-Cnl 2 ) =11. 3 x 10 -15 eV-Cnl 2 " =1. 018 Hence = 1. 292 Mev E1 The Rutherford cross section then beconles R = 3. 600 barns/steradian which in the laboratory systenl is da"(Ep a) dh =, ( d,QCM dh aE(Ern) + f3E(E Ns n L 20 ) jR= 3.617 barns/steradian = 22.10 x 10 -15 eV-Cnl 2 =1 for pure copper (an average nlass of 63.55 a. nl. u. was assunled) -57N :;: 9. 361 x 10 4 counts (corrected for dead time of apparatus) Hence ZeR 2CVO ):;: N S n (o.f(E L ZeR (zeva L ):;: ) + l3E(E lB ) ) N 20 (1.252 x 10- 6 )(3. 617 x 10- 24 ) 1(22.10 x 10- 15 )(9. 361 x 10 4 ) The energies EIB and E 20 for the Li :;: 2.189 x 10- 9 6 scattering were also determined from equations 28 and 30. EIB :;: 1. 298 Mev E 20 :;: 0.9527 Mev ' SL :;: 81 o 13; O. 7510 0.:;: 13:;: 1 ; SCM:;: 90 o 45 ' dOL dTI: :;: 0.9634 CM From the measured stopping cross section of protons in lithium (figure 12) f(E E (E 1] 1B ) "" 2.10 x 10 -15 20 ) "" 2. 67 x 10 -15 ev-cm 2 ev-cm 2 =1. 271 El :;: 1. 287 Mev The Rutherford cross section is R :;: 0.0374 barns/steradian o.E(E lB ) + l3E(E 20 ) :;: 4. 247 x 10- 15 ev-cm 2 -58- Ns =1. 007 (from the spectroscopic analysis given by n the supplier of the Li) = 7.900 x 10 3 counts Background = 490 counts N ave The yicld become s N = 7.410 x 10 3 counts Equation 19 then gives for the laboratory scattering cross section for protons with laboratory energy E (Lab) = 1. 287 Mev from 1 6 0 Li nuclei at a center-of-mass angle of 90 45' dO'") _ (1. 007)(2.189 xl0- 9 )(4. 247 x 10- 15 )(7.410 x 10 3 ) an L - (0.9527 x 10 6 ) = O. 07281 barns/steradian From equation 24 the center-of-mass cross section becomes ~) :: (0.9634)( O. 07281) :: O. 07015 barns/ steradian CM The ratio of the elastic scattering cross section to the Rutherford cross section is then (0.07015) :: (0.03744) :: 1. 874 E (Lab) :: 1. 287 Mev 1 SCM:: 90 o 45 ' -59APPENDIX II Determination of S-wave Scattering Amplitudes An example of the scattering analytii& will be given for the case of El(Lab) :: 1. 879 Mev. Since the method used is described in Section III-c only an outline of the procedure with definite numbers will be given here. A program was written so that most of the cal- culations could be made on the Burrough's 220 computer. An attempt to fit the data with s-wave protons alone was made first by computing the values of the slopes of the six lines intercepts. Y:: A( e)X + B( e) and the six The input parameters for this part of the program are (see Tables 1 and 5) Laboratory incident proton energy. El Center-of-mass s-wave integrated reaction cross section IJ"R Center-of-mass angle eCM Differential elastic scattering center-of-mass cross section dO"( ell em The computer calculates equation 99 which gives A(e) and equation 101 which gives B(e). In addition the computer calculates -B(e)/A(e) and two other quantities b(e) and c(e) defined by the following equations: b(e) :: cos ;(e) (125 ) k(R(e) )1/2 erR c( e) :: 1 - 411'R( e) (126) The quantity -B( ellA( e) is useful in plotting the straight lines -60and the quantities b( a) and c( a) are helpful in computing the ratio of the measured elastic scattering cross section to the Rutherford scattering cross section which is now given (for s-waves) by da"( a) an R(9) = b(a)[A(a)(X-l) - Y] + c(a) (127) For the case of El(Lab) = L 879 Mev these parameters were computed to be (from table 5, (TR(s-wave).:::. o. 068 barns . ) A B -B/A b c -0.6424 -1. 327 -2. 066 1. 796 O. 8656 -1. 265 -0.8757 -0.6921 2.847 o. 6919 110 48 -1. 841 -0.6855 -0. 3724 3.881 0.4485 126 0 6 ' -2. 224 -0.7999 -0.3596 4.579 0.2413 140 53 - 2. 527 -1. 343 -0.5314 5.128 0.0530 159 0 7' -2.784 -2.079 -0. 7466 5.590 -0. 1238 aCM , 0 70 41 , 0 90 45 , 0 0 , The six lines were then plotted in the complex plane and are shown in figure 47. Since they did not intersect at some point within the circle (1 - U)1/2 within experimental error, it was assumed that the resonance at this energy was not an s-wave resonance. Resonant parameters for an assumed p-wave 5/2- state were then estimated by the procedure given on page 45 and resulted in the following set of values at the energy E (Lab) 1 r(CM) p r( CM) =0.834 = O. 872 =1. 879 Mev (see Table 5) -61- A program was also written for the computer to calculate the quantities f 5 / Z' g5/Z' s, G( 9), and H(9). 109. 104. 105, and 106. These are defined in equations 108, The resulting values are f 5 / Z = -0.888 g5/Z = -0. Z18 S = 1. 009 9 G H ZG -0.4604 1. 790 O. ZOZ6 O. OZ636 1. 944 -0.01l6 o. 9389 Z.537 - O. 413 1. 816 3.483 -0.799 140 53 Z.666 4.688 -1. 173 159 0 7' 3.494 6. 100 -1. 537 CM , 0 70 41 , 0 90 45 , 0 110 48 1z6 0 6 ' 0 , The above intercepts H(9) were then added to the s-wave intercepts to give the dashed lines in figure 47. there is still no intersection and the intercepts s-wave, p-wave interference ZG( 9) must be added to the intercepts B( 9) + H( 9). Various values of Z Z It is evident that were tried with the result that only the choice = -(0.44.:!:. O. OZ) would yield an acceptable solution for the s-wave scattering amplitudes. for this value of Z. The values of ZG( 9) are also included above These additional intercepts were added to the B(9) + H(9) intercepts to give the lines plotted in figure 49. The choice -6z- of Z was m.ade sim.ultaneously with the choice for the com.plex point (X, Y) and the graphical solution of equations 121 and lZZ for the com.plex point (f 3/z' g3/z) and the com.plex point (fl/z' gl/z)' Most solutions for the s-wave scattering am.plitudes were discarded because they gave a com.plex point for (f l/z ' gljZ) outside the unit circle. An additional restriction on the above choice was that all three com.plex points be reasonable extrapolations of the values found at lower and higher energies. X = O. 8Z The numbers actually obtained were Y = O. OZ £3/z = 0.85, g3/z = -0. Z9 \12 = O. 77 gl/z = O. 63 { ~ [(l_U)_(XZ+y Z)] }l/z = O. 31 The angular distribution was then calculated from. the equation dcr( 9) ~ R(e) = b(9)[A(9)(X-l)-Y + H(9) + ZG(9)] + c(9) which gave for El(Lab) 9 0 CM ~/R dO" <inexp IR , (lZ8) =1. 879 Mev the following ratios 0 , 0 , , 70 41 90 45 110 48 lZ6°6' 140 53 159 0 7' 4.6Z 6.78 9.90 14. 3 ZO. 3 Z8.1 4.37 6. 80 10. 3 14. 1 ZO.O Z7. 1 0 The second line gives the experim.entally deterrn.ined ratios which are to be com.pared with the calculated ratios. -6Za- APPENDIX III Consideration of the possible l/Z- and 3/Z- assignm.ents for the 1. 84 Mev p-wave If the single assigmnent J'Ir resonance p-wave resonance assum.ed in the analysis has the =l/Z- or 3/Z-, the state m.ay be form.ed through either spin channel and a channel spin =ixing parameter must be introduced. The scattering am.plitude for the l/Z- assignm.ent then becom.es i, j = l/Z or 3/Z where a~/Z is the probability of form.ing the (lZ9 ) l/Z- p-wave state through the l/Z spin channel and a;/z is the probability of form.ing the l/Zp-wave state through the 3/Z spin channel. Since the a~I are proba- bilities and there are only two spin channels, the following relation m.ust hord (130) Since there is only one additional quantity .involved, one speaks of the channel spin =ixing param.eter Z equal to al/ " Z M which is defined here to be The f and g are defined as in equations 108 , and 109. The scattering cross section is then found from. the am.plitude m.atrix in the same way as that described for the case of J'Ir = 5/Z-. of the term.s for a J'Ir All =l/Z- p-wave state m.ust be considered including the channel spin flip term.s (i. e. scattering events which involve a change of channel spin). The correctness of the p-wave resonant term. in the cross section m.ay be checked by integrating the cross section over solid -6Zbangle. The interference terms drop out and the result must agree with the Breit- Wigner formula at resonance r ""Scatt. (R es. ) = Z ZJ+l 4 ... (_p ) ~ kZ r (131) The scattering cross section for a single J'" = l/Z- p-wave resonant state is found to be do- cID (a,E) 1 -fR U =R + (-fR. - r sm g - - Z )(X-l) - cos gy - Z Zk 1 k Z 4k Z + - - Z [ (f l / Z- - 1) + gl/Z-) (13Z) lZk The coefficient Z is defined similarly to equation 103 but now (133) The coefficient S is defined as in equation 104 with the index 5/Z replaced by l/Z-. The s-wave l/Z channel spin parameters, gl/Z' may be eliminated from the equation for Z and by using equations 94. This then gives a straight line equation in terms of the s-wave 3/Z spin channel parameters, f 3/ and g3/Z' which has the same form as Z equation lZZ with the same slope but different intercept. _ S f + Z + S - 3M(SX + Y) g3/Z - 3/Z 1 - 3M (134) -6Zc- This equation combined with the circle equation lZl must then yield an acceptable solution for the correct. s-wave scattering if this choice of J1I" is The p-wave resonant parameters may be found from the 6 3 Li (p, a)He reaction cross section by the procedure given on page 45. An arbitrary s-wave background was chosen which was similar to that used for the J 'II" =5 Z case. /- With the assumption that only s- and p-wave contributions are present, the assignment data. ill' =l/Z- is excluded by the experimental This is primarily due to the fact that the p-wave intensity term is isotropic for J =liz, and this restriction makes it impos sible to fit the observed angular distributions. The analysis in terms of a 3/Z- p-wave state proceeds in exactly the same manner as that for the l/Z- case. The scattering amplitude is given by i, j The channel spin mixing param.eter M above. = 1/ Z or 3/ Z (135 ) is defined in the same way as The scattering cross section then becomes dodn (a,:E) . i; - -::::-2 1 .fR U = R t ( k.fR sm )(X-l) - k cos i; Y - 4k2 Zk i( -i .f: {(£3jz_-l)sin[Z( TJ l-TJ o )-i;] tg 3 /Z_cos [Z(TJl-TJo)-i;]} cos e 1 t 300k Z [13 t 3(50-I3)cos e [(f 3 / Z- - 1) t g3/Z-] + where z:2)[ (f 3 / Z-- l)sin Z(TJl-TJo) t g3/Z-cOS Z(TJl-TJ o )) cos e Z ] Z Z (136) -62df} Z = 34 + 12M _ 81M 2 is defined as in equation 133. (137) The straight line equation for the 3/2 channel spin s-wave parameters is given by equation 134 and the circle equation is given by equation 121. Various values of channel spin mixing parameter M were chosen and an analysis carried out in the same manner as for the 5/2case and the 1/z- case. The case of M =1/z (i. e. if the 3/Z- state is formed equally through both spin channels) may be excluded since it gives an almost isotropic p-wave intensity contribution to the scattering. However, for the cases of the 3/Z- state being formed almost entirely through either the 1/2 spin channel or the 3/Z spin channel, a fit to the data can be made with a different choice of s - wave background. In these cases the s-wave background does not show resonant behavior in the l/Z channel spin but may indicate resonant behavior in the 3/Z spin channel at a mu~h higher energy. -63- REFERENCES 1. F. Ajzenberg-Se1ove, T. Lauritsen, 1959, Nuclear Physics, 11, 1. 2. S. K. Allison, 1958, R. M. P. 30, 1137. 3. M. Bader, R. E. Pixley, F. S. Mozer and W. Whaling, 1956, P. R. 103, 32. 4. K. Bardin. 1961, Ph. D. Thesis, California Institute of Technology. 5. S. Bashkin and H. T. Richards, 1951, P. R. 84, 1124. 6. S. Bashkin and R. R. Carlson, 1955, P. R. 97, 1245. 7. D. Bohtn, 1951, "Quantum Theory, " Prentice-Hall Inc., New York, p. 537. 8. A. B. Brown, C. W. Snyder, W. A. Fowler and C. C. Lauritsen, 1951, P.R. 82,159. 9. W. E. Burcham and J. M. Freeman, 1950, Phil. Mag. 41, 921. 10. R. F. Christy, 1956, Physica XXII, 1009. 11. E. R. Cohen, 1949, Ph. D. Thesis, California Institute_of Technology. 12. W. A. Fowler, C. C. Lauritsen and T. Lauritsen, 1947, R. S. L 18, 818. 13. F. Gabbard, 1961, Private communication to T. Lauritsen. 14. W. D. Harrison and A. B. Whitehead, 1961, B. A. P. S. 11-6, 505. 15. D. R. Inglis, 1953, R. M. P. 25, 390. 16. C. H. Johnson, H. B. Willard and J. K. Bair, 1954, P. R. 96, 985. 17. F. C. Khanna, Y. C. Tang and K. Wildermuth, 1961, P. R. 124, 515. ---- 18. C. C. Lauritsen, T. Lauritsen and W. A. Fowler, 1941, P. R. 59, 241. 19. J. B. Marion, G. Weber and F. S. Mozer, 1956, P. R. 104, 1402. 20. J. B. Marion, 1957, Nuclear Physics, 4, 282. -64- 21. J. B. Marion, 1961, R. M. P. 33, 139. 22. J. A. McCray and P. Smith, 1962, to be published. 23. F. S. Mozer, 1956, Ph. D. Thesis, California Institute of Technology and F. S. Mozer, 1956, P. R. 104, 1386. 24. L. D. Pearlstein, Y. C. Tang and K. Wildermuth, 1960, Nuclear Physics ; 18, 23. 25. S. Rosenblum, 1928, Ann. de Phys. 10, 408. 26. G. A. Sawyer and J. A. Phillips, 1953, Los Alamos Report 1578. 27. W. T. Sharp, H. E. Gove and E. B. Paul, 1955, A. E. C. L. No. 268, "Graphs of Coulomb Functions, " Chalk River, Ontario. 28. C. W. Snyder, S. Rubin, W. A. Fowler and C. C. Lauritsen, 1950, R. S. L 21, 852. 29. R. G. Thomas, 1951, P. R. 81, 148. 30. J: B. Warren, T. K. Alexander and G. B. Chadwick, 1956, P. R. 101, 242. 31. W. D. Warters, 1953, Ph. D. Thesis, California Institute of . Technology, p. 37. 32. W. A. Wenzel, 1953, Ph. D. Thesis, California Institute of Technology, p. 54. 33. W. Whaling, 1958, Handbuch der Physik, Bd. XXXIV, p. 193. TABLE 1 DIFFERENTIAL ELASTIC SCATTERING CROSS SECTION (Page 28) E1(Lab) (Mev) 0.495 dO" em 544 252 152 114 88.8 78. 7 0.95 0.99 1. 06 1. 10 1. 06 1. 14 0.792 70°41 ' 90°42' 11 0°28' 125 0 2?' 140°2 , 159°32 255 119 77.5' 62.9 53. 3 42.9 1. 12 1. 20 1. 40 1. 54 1. 65 1. 59 1. 087 70°41: 9 00 42, 110°28 125 0 2?' 140 0 2 159°32' 138 77. 9 60.5 51. 3 46. 2 42. 2 1. 14 1.48 2.05 2.38 2.69 2.95 70°41 : 90°42 110°28', 125 0 2? 1. 15 1. 29 1.51 1. 70 1.92 1.92 70°41 ' 90°42' i~~~;2' 208 101 66.5 54.8 49. 1 40.5 1. 186 i~~~~2' 0.97 1. 06 1. 24 1. 22 1. 26 1. 32 0.892 i i~~~;: 398 187 123 87. 2 73.0 63.0 108 73.7 60.4 54.5 50. 1 45.3 1. 07 1. 67 2.43 3. 01 3.47 3.73 70°41 ' 90 0 42' 110°28' 125 0 2?' 140°2 159°32' 308 143 85.0 69. 2 56.4 47.0 1. 03 0.989 70°41 ' 90°42' 110°28' 125 0 2?' 140°2 159°32' 164 87.9 63.2 52. 8 46.8 42.5 1. 12 1. 39 1. 77 2.03 2.26 2.45 1. 286 96. 1 70. 2 58. 8 56.5 56. 3 55. 6 1. 12 1. 87 2.82 3.70 4.62 5.38 70°56: 90°42, 110°28, 12502~ 140°2 159°32' 0.594 0.692 dO" ~/R SCM (degrees) 70°41 ' 90°42' (m.b) 1.11 1. 17 1. 31 1. 32 1. 32 E1(Lab) SCM (Mev) (degrees) ern dO" /R E1(Lab) dO" SCM <in (Mev) (degrees) (m.b) ~/R <in (m.b) g~~~?: 140°2 159°7' 70°41' 90°45' 11004~' 126°6 , 140°53 159°7' , 0"- , U1 TABLE 1 (Cont.) dO" E1(Lab) SCM (Mev) (degrees) em ~/R (rob) E1(Lab) SCM (Mev) (degrees) dO" em ~/R (rob) em ~/R dO" E1(Lab) SCM (Mev) (degrees) (rob) 1. 386 70 0 41' 90°45' 110 0 48' 126°6' 140°53' 159°7' 90.6 70.5 63.9 65.4 70. 6 71. 2 1. 22 2. 18 3.56 4.98 6.72 8.06 1. 681 70°41' 90°45' 11004~' 126 0 6 140°53' 159°7' 151 116 111 120 138 157 2.98 5. 28 9.10 13.5 19.4 26.0 1.976 70°41 ' 90 0 45' 110 0 48' 126 0 6' 140°53' 159°7' 167 106 88.5 82; 6 88.9 104 4.57 6.67 10. 0 12. 8 17. 3 23.9 1.485 70°41' 90°45' 110°48' 126°6' 140°53' 159°7' 96. 2 80. 8 75.7 82.8 90.4 98.8 1.49 2.87 4.84 7.24 9.89 12. 8 1.780 70°41' 90°45' 11 004~' 126 0 6 140 0 53' 159°7' 174 124 115 121 134 154 3.86 6.34 10.5 15. 2 21.0 28.6 2.076 70°41' 90°45' 110°48' 126 0 6' 140°53' 159°7' 152 95.5 74.7 69.5 74.2 83.2 4.56 6.60 9. 31 11.9 15.9 21. 1 1.585 70°41 ' 90°45' 110°48' 126°6' 140 0 53' 159°7' 113 96.4 94.8 1{)3 117 134 2.00 3.90 6.89 10. 3 14.5 19. 8 1.879 70°41' 90°45' 110 0 48' 126°6' 140°53' 159°7' 177 119 101 101 114 130 4.37 6.80 10. 3 14. 1 20.0 27. 1 2.174 70°41 ' 90°45' 11 0°48' 126 0 6' 140°53' 159 0 7' 139 87. 2 64.5 58.4 61. 2 67.3 4.60 6.63 8.83 11.0 14.4 18. 8 , , 00- TABLE 1 (Cont. ) E (Lab) SCM 1 (Mev) (degrees) 2.273 70°41 ' 90 0 45' 110 0 48' 126 0 6' 140 0 53' 159°7' dcr em ~/R (mb) 130 79. 8 57.4 52.3 51.4 56.4 4.70 6.63 8.58 10. 6 13. 2 17. 2 E (Lab) SCM 1 (Mev) (degrees) 2.570 70°41 ' 90°45' 11 0 0 48' 126 0 6' 140 0 53' 159°7' dcr em ii/R 106 69.6 50. 1 44.5 41.9 45.4 4.90 7.39 9.57 10. 8 12. 7 16. 3 (mb) 2.372 70°41 ' 90°45' 110°48' 126 0 6' 140°53' 159 0 7' 119 76. 1 55.5 47.8 45.2 50.2 4.69 6.88 9.05 10. 7 12. 6 16. 7 2.664 70°41 ' 90 0 45' 110 0 48' 126 0 6 ' 140 0 53' 159°7' 104 67.9 49.6 41. 1 37.7 38.2 5. 17 7.78 10. 2 11.6 13. 3 16. 0 2.470 70°41 ' 90°45' 110 0 48' 126 0 6' 140°53' 159°7' III 4.73 7. 10 9.28 10. 8 12. 7 16. 3 2.762 70°41 ' 90 0 45 ' 110 0 48' 126 °6' 140 0 53' 159°7' 100 66.2 48.8 40.2 36.0 36. 1 5.37 8.14 10. 8 12. 0 14.3 16. 2 72.4 52.5 44.5 41.9 45.4 E (Lab) SCM 1 (Mev) (degrees) 2.861 70°41 ' 90 0 45' 110°48' 126 0 6 ' 140 0 53' 159°7' Qcr dO ii/R 96.7 64.7 49.2 39.9 35. 3 34.8 5. 5 ~ 8.52 11.7 13. 0 14.4 16. 7 (mb) I a- , ~ -68TABLE 2 6 PROBABLE ERROR IN THE Li SCATTERING CROSS SECTION (Page 30) ESTIMATED PROBABLE ERROR (%) RELA TIVE ERROR 1. Resolution to solid angle ratio Uncertainty in factor aE(E lB ) + I3€(E 20 ) due to O. 50 uncertainty in angle and 2"/0 relative uncertainty in € Cu 1.4 Statistical uncertainty in Cu yield 0.5 Fluctuations in firing voltage of current integrator 0.5 Fluctuations in spectrometer energy calibrations O. 3 Uncertainty in Cu Rutherford cross section due to 0.5 0 error in angle and O. 5% err or in EI Back angles Forward angles 1.0 1.9 Uncertaintl in factor aE (EIB) + j3€ (E 20) due to O. 5 uncertainty in angle and 2"/0 relative uncertainty in € Li 1.5 3. Statistical uncertainty in Li,6 yield 1.5 4. Fluctuations in firing voltage on current integrator 0.5 Uncertainty in target composition Ns/n O. 2 2. 5. Relative probable error in cross section All angles -69- TABLE 2 (Cont.) ESTIMATED PROBABLE ERROR ( %) ABSOLUTE ERROR 1. Re solution to solid angle ratio 3. 5 Uncertainties are the same as for the relative error with the exception of the factor a€ (E IB ) + f3€ (E 20 ) which is uncertain by about 3% on an absolute scale. (Bader, 1956) 2. Uncertainty in factor for Li a € (E IB ) + f3€(E20) 3. 0 3. Statistical uncertainty in Li 6 yield 1.5 4. Fluctuations of firing voltage of current integrator 0.5 Uncertainty in target composition Ns/n O. 2 5. Absolute probable error in cross section All angles - 5% -70TABLE 3 COMPLEX SCATTERING AMPLITUDES (Page 52) E (Lab) 1 f 1/ 2 gl/2 f 3/ 2 g3/2 0.495 0.594 0.692 0.792 0.892 0.989 1.087 1. 186 1.286 1.386 1.485 1.585 1. 681 1.780 1.879 1. 976 2.076 2.174 2.273 2.372 2.470 2.570 2.664 2.762 2.861 0.981 0.973 0.972 0.986 0.939 0.977 0.970 0.962 0.946 0.934 0.932 0.891 0.900 0.874 0.770 0.629 0.315 0.143 -1).058 -0. 235 -0.400 -0.470 -0.580 -0.650 -0. 688 -0.035 -0. 065 -0.068 -0.178 -0. 106 - O. 217 -0.240 -0.238 -0. 215 -0. 162 -0.160 -0. 038 0.070 0.424 0.630 o. 779 0.950 0.987 0.945 0.880 0.788 0.710 0.567 0.465 0.324 0.981 0.973 0.972 0.944 0.982 0.930 0.927 0.928 0.930 0.934 0.932 0.950 0.932 0.880 0.850 0.810 0.798 0.770 0.774 0.760 0.750 0.727 0.710 0.700 0.673 -0.035 -0.065 -0.068 -0.090 - 0.200 - O. 130 - O. 133 -0. 132 -0.140 -0.162 -0.160 -0.215 -0.240 -0.273 -0.286 -0.318 -0.323 -0.370 -0.400 -0.440 -0.455 -0.507 -0.540 -0.580 -0.600 - 71- TABLE 3 (Cont. ) E (Lab) 1 X 0.495 0.594 0.692 0.792 0.892 0.989 1.087 1. 186 i.286 1.386 1.485 1.585 1. 681 1.780 1.879 1.976 2.076 2.174 Z. 273 2. 372 2.470 2.570 2.664 2.762 2.861 0.981 0.973 0.972 0.960 0.954 0.947 0.942 0.940 0.938 0.934 0.932 0.928 0.922 0.880 0.820 0• .75-.0 0.640 0.560 0.500 0.430 0.370 0.330 0.280 0.250 0.220 Y Z -0.035 -0.065 -0. 068 - O. 120 -0.135 ~O. 155 -0.165 -0. 165 -0. 165 -0.162 - o. 160 -0.156 -0. 135 -0.040 0.020 0.050 0.100 0.085 0.050 0 -0.040 - O. 100 -0. 170 -0. 230 -0. 290 0 0 0 0 0 0 0 0 0 0 0.124 0.740 -0.550 -0.480 -0.440 -0.433 -0.425 -0.450 -0.450 -0.470 ...{}. 490 -0.534 -0.567 -0.596 -0.614 £5/2 g5/2 0.999 0.998 0.995 0.992 0.982 0.964 0.938 0.887 0.810 0.659 0.413 0.031 -0.468 -0.844 -0.888 -0.704 -0.461 - 0.285 -0.088 0.035 - O. 142 0.209 0.266 0.317 0.353 0.034 0.061 0.093 O. 121 0.175 0.253 0.332 0.443 0.565 0.725 0.875 0.947 0.798 0.334 - O. 218 -0.600 -0.818 -0.905 - 0.953 -0.963 -0.958 -0.949 - 0.938 -0.924 -0.912 -72- TABLE 4 RESONANT PARAMETERS (Page 54) Nucleus Be 7* Be 7* Be 7* Li 7 * (3/2+) (1/2+) 5/2- 5/2- Mev 0.77 2.76 1. 84 0.262 Mev 1. 03 1. 23 0.836 0.154 7.58 7.70 (Units) J'lr ER(Lab) r(E R ) EA. r (Mev above ground) (E ) Mev p,n R 2 Yp , n Mev-f 2 ep,n 0.025 0.90 0.798 O. 118 O. 17 1. 82 5.02 4.85 O. 01 O. 10 0.2 8 0.26 ro.(E ) R Mev 1. 00 0.33 0.038 0.036 2 Yo. Mev-f 1.7 0.25 O. 101 0.091 0.20 0.03 O. 012 0.012 ea.2 -73TABLE 5 ASSUMED S-WAVE INTEGRATED REACTION CROSS SECTION AND P-WAVE RESONANT PARAMETERS (Page 59) E (Lab) 1 (Mev) 0.495 b.594 0.692 0.792 0.892 0.989 1.087 1. 186 0.286 1.386 1.485 1.585 1. 681 1. 780 1.879 1.976 2.076 2.174 2.273 2. 372 2.470 2.570 2.664 2.762 2.861 CJ"R (barns) 0.068 0.069 0.070 0.070 0.070 0.069 0.067 0.065 0.063 0.063 0.061 0.060 0.058 0.059 0.068 0.072 0.072 0.073 0.076 0.079 0.080 0.082 0.083 0.084 0.084 ER(CM) r(CM) r (CM) (Mev) (Mev) (Mev) 1. 986 1.960 1.935 1.923 1.901 1.872 1. 851 1.823 1.796 1.760 1.723 1.680 1.639 1.595 1.559 1.530 1.501 1.491 1.455 1.438 1.422 1.411 1.402 1.393 1.390 0.076 O. 113 0.150 0.176 0.228 0.292 0.345 0.411 0.468 0.539 0.605 0.660 0.733 0.777 0.872 0.923 0.990 1.054 1. 123 1. 190 1. 242 1. 317 1.377 1.439 1.504 0.053 0.089 0.126 0.151 0.201 0.265 0.316 0.381 0.437 0.507 0.572 0.625 0.697 0.740 0.834 0.884 0.950 1. 013 1.080 1.146 1. 197 1. 271 1.330 1.390 1.454 p -74aFigure 1 Thick Target Diagrams (Page 5 ) Figure a represents the target beam geometry for a typical situation. 8 1 The perpendicular to the target surface is oriented at an angle from the direction of the incident beam and at an angle 8 direction of the exit beam.. Z from the Particles which strike the front surface of the target with bombarding energy EIB scatter or form a reaction at the surface and leave at some angle 8 Z with an energy E ZB " The mag- netic spectrometer may be adjusted to accept particles which originate only within a certain lamina ~s in the target" The energy of a particle before a scattering or reaction event is given by El and that after the event by E " Z Since some energy is lost by the particles in traveling from the surface to the lamina. El will be less than EIB and similarly the energy of the particle accepted by the spectrometer E ZO will be less than E Z" Figures band c represent the scattering situation in the determination of the stopping cross section for protons in lithium.. In the first case the target is just copper; in the second. copper plus a thin layer of lithium.. -74 - o. b. c. -75a- Figure 2 Spectro=eter Profile of Copper Target (Page 13 ) The voltage across a precision resistor is plotted on the abscissa and is proportional to the current through the flux=eter in the m.agnetic spectrom.eter. As indicated on page 13 this current is inversely pro- portional to the square root of the energy of the particles accepted by the =agnetic spectrom.eter. The yield or num.ber of particles counted for a given charge collected is plotted on the ordinate. At a given angle the protons scattering elastically from. copper will have a m.axim.um. energy corresponding to those scattered fro= the front surface of the target. Most of the protons will penetrate into the target, scatter at som.e point and also com.e out in the sam.e direction as the protons scattered from. the front surface. These, however, will lose energy while passing through the target and will be recorded at a lower energy. The result then is a step function with the front edge at the calculated m.axim.um. energy. This copper edge was used to calibrate the =agnetic spectrom.eter i. e. to find the constant C MS in equation 30. The incident proton energy was about 1. 3 Mev and the laboratory angle 81 o 13 ' . -75- Cu(p,p)Cu 5 • • • 4 3 ~ (f) ~ . z ::> 0 u 2 o • L -_ _ _ _ ~ _ _ _ _- L _ _ _ _ .562 ~ ______ ~ ____ .560 FLUXMETE R ~~--~--J .558 (volts) -76aFigure 3. Spectrometer Profile of Lithium Target (Page 13 ) This profile was taken with the 10. 5-inch spectrometer and a Li target in the 2. 5-inch scattering chamber. The 0 16 and C 12 peaks correspond to protons which are scattered from the thin oxygen and carbon contamination layers on the surface of the lithium. of the Li 6 profile is given by equation 18. The shape The incident laboratory ' proton energy was about 0.80 Mev and the laboratory angle 116 o 54. 6 II) o N u o o(l) 0 - V > (I) II) -l 0 (l) a. ~ - a:: w -l W a. II) ~ ~ x ::::> II) 0 -.J (l) (l) LL • -l o C\J (J) 0 0 lO q r<1 C\J C\J lO ~ 'SlNnO:) 0 t.D lO OCT! -77aFigure 4 Spectrometer Profile of Lithium Target (Page 14) This profile was taken with the 10. 5-inch spectrometer and a Li target in the 2. 5-inch scattering chamber. of the Li 6 sample was only 95.7%. the Li However. since the purity 7 profile is also present. An indication of the age of a target is given by the "tail" on the and 0 16 peaks which represent diffusion into the lithium. e 12 The incident laboratory proton energy was about 1. 00 Mev and the laboratory angle 1 155 0 34 • 6 CD o N U 0 <t CD - I'-; -I a. a. ~ - CD -I (J) 0 +- CD I'- 0 > ~ W IW ~ x ,... -I CD 0 ::J N CD LL o U) CD o o (]) o co N )l 'SlNnOJ o ...J -78aFigure 5 Spectrometer Profile of Lithium Target (Page 21) This profile was taken with the l6-inch spectrometer and a Li target in the 6-inch scattering chamber. The 0 16 and C 12 6 peaks here. however. are much reduced because of the improved vacuum in this chamber. The Li 6 edge also appears to be sharper than the Li 6 edges found with targets in the 2. 5-inch scattering chamber. The incident laboratory proton energy was about 1. 60 Mev and the laborao I tory angle 81 13. -78- . ..J --•,. o )I • S.LNnOO -79aFigure 6 Spectronleter Profile of Lithiunl Target (Page 24) This profile was taken with the l6-inch spectronleter and a target in the 6-inch scattering chanlber. Li 6 The solid dots represent the profile taken just after the target was nlade and show very little " di cat"10n 0 f 1n e 12 and 0 16 cont anllna " t"100. enough to stop a. ++, s and He 3++' s When a foi1~ which is thick but not protons at this energy,. is placed in front of the detector •. the profile indicated by the crosses is found. This then is taken as evidence that the background consists nlainlyof a. ++, s and He 3++' s. The profile represented by the circles was taken twenty-four hours later and gives an indication of the quality of the target used and the vaCUUnl in the 6-inch scattering chanlber. The incident laboratory proton energy was about 2.3 Mev and the ' laboratory angle 81 o 13. -79 - o N ~ ID -.J a. a. ~ - 0 <:t- ID 'It. -1 VI .0 > ---- 0 <.D ~ a:: w ~ w ~ x ~ -.J u... 0 ID CO -1 ~ o o l{) o N o >l 'SlNnOJ l{) . -80a- Figure 7 Resonant Gamma-ray Yield From the F The 80 0 19 16 . (P. al')O Reaction (Page 15 ) electrostatic analyzer was calibrated by using the known 872.5 kev resonance in the F 19 (p. al')016 reaction. The dots represent the curve taken from left to right and the crosses represent the curve taken immediately afterwards from right to left. - 80 - F 19 ( p,a r) 0 16 E =0.8725 MEV 20 15 ~ (J) IZ => o u 10 5 .860 o .870 ELECTROSTATIC .880 ANALYZER (volts) -8laFigure 8 Neutron Threshold for the Reaction Li 7(p. n)Be 7 (Page 16) In this graph the two~thirds power of the number of neutrons counted for a given charge collected is plotted against the setting of the electrostatic analyzer potentiometer. The analyzer is calibrated by using the straight line extrapolation near the threshold and the well-established threshold energy of 1880. 7 + 0.4 kev (Marion~ 1961). 700 200 100 0' .820 .822 .' ELECTROSTATIC • .824 • ..... ' .826 ANALYZER .828 (VOLTS) .830 -82aFigure 9 Scattering Geo=etry of 2. 5-inch Cha=ber (Page 10) The incident beam comes into the chamber at approximately 10 0 above the horizontal plane and leaves the chamber at approximately 10 0 below the horizontal plane. The scattering angle the relation . 6L slnT = cos (1 . 6H Sln T 6 L is given by -82 - -- - -I / GEOMETRY OF THE ILLUSTRATING SCATTERING PROCESS THE RELATIONSHIP OF THE HOR IZONTAL PLANE AND THE BEAM PLANE -83aFigure 10 The 6-inch Target Chamber and Furnace (Page zq This drawing illustrates the relative positions of the scattering chamber and furnace. This arrangement with the cold trap separating the two regions was found to be a very satisfactory way of obtaining a clean copper calibration target and a lithium target in the scattering chamber at the same time. -83- TA.'(T .OD ______ ~ VII"W'''' OU ...TZ /'----____ T•• I[T ~OlO(R TAIllIIT TAltI[T COLD TAU -84a- Figure 11 Stopping Power Measurement Spectrometer Profile (Page 21) The curve on the right represents the copper edge resulting from the scattering of protons from a clean copper surface. The two curves on the left represent the copper edges resulting from the scattering of protons from a copper surface upon which a thin layer of lithium has been deposited. The displacements between the latter two edges and the clean surface copper e dge are due to the energy loss of the protons in the lithium layer. The incident laboratory proton energy ' was about 2.0 Mev and the laboratory angle 90 o 44. -84- -a:: lIJ ~ ~ . L- >0 ...J ...J -,&; 'j ~ (,) ...... 0- 0. ~ (,) ~ o o CD __ ~~ __ ~~ __ ~ ____ ~ ____ ~ ____- L____ ~ 0 0 0 0 0 <t S.l.Nno:> ____ ~ __ o ~O 10 -85aFigure 12 Atomic Stopping Cross Section for Protons in Li (Page 24) The solid points are measurements of Bader et al., (Bader, 1956) and Warters et ale (Warters. 1953) of the stopping cross section for protons in Li. The solid points with vertical bars are determined from the a.-particle measurements of Rosenblum, (Rosenblum. 1928). The solid curve is the Bloch curve as derived by Whaling, (Whaling, 1958). Relative stopping power measurements made in this experiment are indicated by the crosses and triangles and show that indeed the Bloch curve applie s to lithium as well as to all of the other elements (Whaling, 1958). ATOMIC STOPPING CROSS SECTION FOR PROTONS IN LITHIUM - 15 2 10 ev~ cm .. E , .. ..' 0 u > r ~~ + ++ l~ u... .I:;L ~ w 1&1 .15 .3 3.0 .5 E, (Mev) 5.0 8.0 -86a- Figure 13 Dead Tin1e Measuren1ent of Scintillation Counter (Page 26) In this graph the difference between the true nUn1ber of counts and the nUn1ber of counts recorded for a given charge collected is plotted against the reciprocal of the tin1e necessary for collection. The counting rate is then varied by increasing the bean1 current of protons which are being scattered fron1 a copper target. An estin1ate for the true nUn1ber of counts is found by counting at a very low counting rate. The dead tin1e is then given approxin1ately by the for=ula _ slope p 2 N where N is the esti=ate for the true nUn1ber of counts and the slope of this curve is used. Here N was found to be p = 10 ~sec. =1. 828 x 10 5 counts and the dead tin1e 24K DEAD TIME .-z 20K 0 ' (f) • MEASUREMENT ::> u z w u z w a::: w 16K ( 12K • SK SLOPE u.. u.. o .. C' '" 5 3 . 41 X 10 CNTS-SEC <J 0'0 8 1'6 2'4 32 lit 40 4'S (X 10- 3 sec-I) 5'6 -87aFigure 14 Excitation Function for 6 6 Li (p. p)Li (Page 27) The ratio of the measured elastic scattering cross section to the cross section for Rutherford scattering from a point charge is plotted as a function of laboratory proton energy (in Mev) for three C. M. angles corresponding approximately to the zeros of the first and second Legendre polynomials and the farthest back angle obtainable. The crosses indicate data taken on the 2-Mev machine and the dots indicate data resulting from measurements on the 3-Mev machine. 29 ....,. THE ELASTIC SCATTERING 27 .' OF PROTONS FROM 25 '. Li 6 NUCLEI 23 8 eM • 159· 7' '. -. 21 19 17 ........... _, 15 , 13 ()) -..j It: ~ ~ ~ , "' - . 1:1 II .' 13 .' 9 II 7 .' 9 5 7 3 5 I 9 8 ' 159· 32' ••: ~ ...... I..! ''II:.~.:.:.:'"''' '' ' ''' ''' 3 0. ..... ... .,. 5 3 8 eM, 90· 42 ' ................................ 1= 0.2 0. 4 0..6 0.8 1.0 .' " ..." ........ , ........ _••• ,_. ,....... ..... .. : ...... . .. 8 eM ' 125· 29' ........... .. ,,' .., .. , 8eM • 126· 6' 8eM ' 90· 45 ' .., ............... , .............. ... .... , ..................., . • - .... , ...................... ,. 1.2 1.4 E, (MEV) 16 18 2.0 22 24 2.6 28 -88aFigure 15 0 6 6 Scattering Cross Section for Li (p. p)Li at SCM= 90 (Page 27) The measured cross section is indicated by the solid dots. The solid curve represents the calculated Rutherford cross section. The curve indicated by the open circles is that calculated for a 5/2- pwave resonant state as shown on page 27. The curve indicated by crosses is the sum of the Rutherford cross section and the cross section due to the 5/2- p-wave resonant state. The triangles represent the s-wave scattering cross section calculated from the s-wave scattering amplitudes which were determined in the scattering analysis of the data. The solid curve through the experimental points is the sum of the s-wave background, the 5/2- p-wave resonant state, and the Rutherford cross section; and represents a theoretical fit to the data. - 88- ... -89aFigures 16 to 40 Angular Distributions for Li 6(p. p)Li 6 Scattering (Page 28) In the following figures the ratio of the measured scattering cross section .to the calculated Rutherford cross section is plotted as a function of center of mass angle. The error marks indicate relative uncertainties and are 30/0 for the backward angles and 40/0 for 90 70 0 • 0 and The curves are calculated from equation 97 using the values of the scattering amplitudes deduced from the analysis and given in table 3 and shown in figure 48. -89 - - I I I I ~ ... I I 0 - U) U) W et: -0 - l{) o -.;t o -I'<) o -N o -- I~ o > W -0 ~ l{) (j) _0 .;t (j) o " CD I- <! _0 ..J CO - W _0 I'- r I I I I I 0 L-~------~-------L------~------L-----~------~OW o ~ l{) N 0 N l{) 0 l{) 0 :::) (,9 LL -90~ T T T H J I 0 - lD I'- W a:: ::) - 0 LO (9 LL o -~ o -r<> o -N -0 i- 0 > -0 W ~ ~ ()) ID ~ ---a..a.. '-' ID -.J - 0()) LO 0 II CD <l: - 000 -.J - w 0 - I'- H I I I ~ N N I I I 0 L---oL-----~LO-------O~----~LO-------O~------LOL-----~OlD 0 ~ U Cb -9 1- 0 <..0 H ro W 0:: :::) 0 L{) <..? lL. o r<l o C\J o ~ u CD 0 0 > W ~ C\J (J) 0 <..0 <D .- 0 -.l " 0.. CD (J) - <t-.J 0 0.. ro ~ <D -.l w 0 I'- L -__L -____ o ~ ~L_ L{) C\J ____ ~ ______ 0 C\J ~ ______ L{) ~ 0 ______ ~ ______ L{) 0 o ~<..O 0 - 92- 0 ~ I--+-i en W 0 ~ o o;t o ~ o N o 0 0 > W ~ N Q) 0 I'-- ID --.J Q. en 0 " - <t al Q. --- - - ID --.J 0 -1 CX) w 0 I'-- ~ __ ~ o ~ ______ ~ _ _ _ _ _ _L -_ _ _ _ ~ 0 N N ~~ ~ ____ ~ ______ 0 ~~ ~ 0 ____ o ~~ 0 n::: => t9 LL - 93 - 0 W 0 N W a:: 0 ~ o <;t o ~ o N o > W 0 0 ~ N en CO <D .- 0 Cl. m -.l Cl. <D -.l en " - <l:--.J - 0 0 CO w 0 r-- o ~~~----~~----~~----~~----~~----~~----~~ o ~ 0 ~ 0 ~ 0 ~ N N 0 :::> <.9 LL -94 N W a::: o ~ o <;;t o rI) o N o 0 0 > W ~ (J) 00 <D (J) --.l 0 .- Cl. Cl. --- <D --.l 0 (J) " c::r CO 0 -l 00 w 0 I'-- L-~ ______- L______- L______J -______ o I"Ii ~ N o N ~ ~ 0 ____ ~~ ~ 0 ____ o ~~ 0 :J (9 LL -95 - 0 to C\J C\J W a:: 0 l() o <;t o r() o C\J o 0 0 > w ~ I'It) --.J Q. (l) 0 0 0"> " (l) 0 <! --- wQ. It) (l) ...J --.J 0 I'- ~~L- o ~ o ____~______~______~______- J_ _ _ _ _ _~~_ _ _ _-JtD l() 0 C\J C\J l() 0 l() d 0 :::::> t.9 ll... -96- 0 {D 1'0 C\J W 0 to o ¢ o 1'0 o C\J o 0 0 > W ~ lD <D ---.J 0 CX) (J) " --- m <t --- -w0. ~ 0. <D 0 -.J CX) -.J 0 f'-- o L-~L-~~~L-~~~L-~~~L-~~~L-~~-J~~~~{D Q {D 0 ~ 000 ¢ 1'0 N 0 0 cr ::> 19 LL -97- 0 (,!) ¢ N W c:r: :::) 0 t9 lO LL o <:t or') o N 0 0 > W ~ 1.0 <D ~ ex) 0 C\J (J) " 0.. - ex) 0.. <! ..J ~ W 0 CO <D 0 I'-- L-~L- o N ____ ~L- 0 0 ____ ~L- CO ____ ~ o ______- i______ ______-J(,!) 0 0 0 <:t N ~ -98- 0 \.0 If) C\J W n: 0 If) :::) (9 LL o <:t o r0 o C\J o 0 0 > W ~ \.0 CD r<'l ID --.J Q. 0 Q) " - <!-.l CD 0 Q. --- w ID --.J CD 0 I'- L -__L -______L -______L -______L -____ Q C\J q o ~~ ____ ~ ______ o o o o CD \.0 <:t N o ~\.O o ~ u CD W N W 0:: o lO o ~ o r0 o N o 0 0 > w ~ lO 0 OJ ~ ID -.J Q. (J) " - <r - Q. ID -.J CD 0 -l OJ w 0 I'- o ~~------~------~------~------~----~~----~W o N 0 000 0 OJ W ~ 0 N 0 ~ 19 LL -100- o lD o ~ o r0 o N o 0 0 > w ~ lD CO <.0 .- lD 0. " en ....J - 0. <.0 ....J 0 (J) <! ...J 0 <Xl w 0 I'- o L---O~----~O~----~O------~O. ------~O------~O------~O~ ~ N d N ~ N <Xl ~ 00 N W o (9 lO LL o <:t o r<> o N o 0 0 > w ~ 00 '"-.J 0.. 0 <..D (]) " - <t 0.. '" -.J 00 0 -.J <D w 0 I'- 0 (!) 0 <D N 0 <:t N 0 0 0 0 0 0 (!) N <D <:t N ~(~~ )/( 3;~) 0 a:: => -102- w en C\J 0 W 0:: => 0 <.!) LO l.L o ~ o rf) o N 0 0 > W ~ 0 a:> 0 I'- to --1 (J) " ,-- a. a. (IJ - <! 0 to --' ---w to --1 0 I'- 0 0 to N 0 ~ N 0 0 N 0 0 W N ~ ( ~~ )/( ~~) 0 to 0 <i W 0 -1030 t.D 0 r0 W 0: 0 L() o <;t o r0 o C\I o 0 0 > w ~ O'l I"<X> ID 0 C1l .-J ---a. -" a. ......... ID .-J CD 0 <t -w CO -.J 0 I'- o O~------O~------~O--------O~------OL-------O~------~O~----~Ot.D 00 C\I ~ C\I 0 C\I t.D C\I 00 ~ ~ (9 lL. -10 4 - 0 lD r0 W a:::: :::> 0 10 o v o r<> o N o 0 0 > W ~ U) I'<D .....J 0.. ~ 0.. ---- en 0 " « 0 (]I CD -l 00 <D .....J w 0 r-- o ~~------~------~------~------~----~~----~lD q V N o o N q 000 lD N 00 V 0 <.9 lJ... -105- 0 <D C\J rO W a:: 0 l{) o <j" o r<"'l o N o 0 0 > w ~ I.D I'- 0 0 <D -1 .--. Q. ~ Q. <D -1 CJ'> N " CD <! -.J 0 CD w 0 t- 0 q 0 q 0 0 \!i 0 <j" N CD <j" N N ~ ( ~~ )/( ~~) 0 0 <D => <..9 lL. -1060 (J) r0 r0 W cr <..9 l() LL o ~ o r<"> o N o 0 0 > W ~ ~ f'- 0 ID (]) -.J N II a.. - CD<! a.. --- -.J --.J w ID 0 <Xl 0 ~ 0 0 <i N (J) 0 0 0 (!) 0 0 0 N <Xl ~ N Cj(UP ...op )/( ~~ ) 0 => 0 -107- a <.D <;t 1"1') W cr: a l[) ::) <.9 LL a <;t a r0 a N a a a > w ~ r0 r'<D .-J ...--Q. (]) " - CD <I: -l .-J w - a N N Q. <D a co a I'- a ~--a~------a~.------~a~----~a~------a~------~a~----~a<.D ~ N a N <.D N ro ~ ~ u Q) -1080 (,!) l{) r0 W a:: ::::> 0 LO o v o r<) o N o 0 0 > W ~ C\J I'- <D r0 0 C\J (j) .-J II 0. en 0. ...J ....-- <! ---.-J w 0 OJ <D 0 r-- 0 0 ~ N 0 0 0 0 0 0 (,!) N OJ V N ~ (~~ )/( ~~) (,!) 0 19 LL -109- 0 f.D to r0 W 0:: 0 LO :::> <.9 LL o C\J o > uJ 0 0 ~ 0 I'<;t ~ 0 C\J (J) -.J ---- " --- 0.. CD 0.. <t ..J 0 co ~ -.J w 0 I"- 0 0 <;tC\J 0 O 0 0 0 0 f.D C\J co <;t- C\J ~ ( ~~ )/( ~~) f.D 0 ~ u Q) -110 - 0 I.D f'r0 W 0:: 0 If) ::> t.? lJ... o ~ orf') o C\J o > 0 0 W ~ 0 f'LO N ID --.J .-0.. ~ 0.. ID --.J 0 (1) II CD <t 0 -1 CO - w 0 I'- L -__L -_ _ _ _ _ _ o ~ 0 ~ 0 C\J C\J ______ ~ ~ I.D ______ ~ _ _ _ _ _ __ L_ _ _ _ _ _ ~ 0 C\J CO o ~------~I.D ~ ~ 0 ~ u CD - I II - 0 to CX) 1'0 W a::: 19 L() lJ.. o N o > 0 0 W ~ <;t \.D \.D III N ---1 II --- 0.. 0 (1) CD <t - w 0.. 0 ~ <X) I II ---1 0 I'- ~~~ Q ~ N ______ ~ Q 0 N ______ ~ ______ ~ 0 Q uj N ______ ~ 0 ______ a:i ~~ 0 o;i ____ o ~tO 0 => 0 -1120 <!) (J) r0 W 0:::: 19 LO LL o <;t o rt') o C\J o 0 0 > W ~ N <.D CD ..--l --- Q. ~ Q. CD ..--l I'- 0 N O'l II CD 4 0 - CO -1 - w 0 r-- o L -__~------~-------L-------L------~------~------~<!) Cl Cl 0 Cl 0 0 <;t C\J 0 <!) C\J CO <;t C\J 0 => 0 - 113- o ¢ w a:: o If) => <..9 LL o v ort') o N o 0 0 > W ~ U) (j) .-J 0 00 N C1'l ---- " 0 (j) -.J CO .-J W 0.. 00 -- 0.. <l: 0 I'- L -__L -______ Q v N ~ ______ ~ _______ L_ _ _ _ _ __ L_ _ _ _ _ _ 0 0 Q 0 ~ N N 0 co ~ ______ Q V o ~~ 0 ~ u CD -114aFigure 41 Spectrometer Profile of Reaction Products (Page 28) This profile was taken with the l6-inch spectrometer and a target in the 6-inch scattering chamber. Li 6 The curve represented by the dots is the measured superposition of the thick target a ++ and He 3++ profiles which result from the 6 3 Li (p, a)He reaction. method used to interpret the profile is given on page 28. The The incident laboratory proton energy was about 2. 3 Mev and the laboratory angle 95 0 45'. -114- + + ,.,., 'd \ \ -. - \ :I: 'd - \ \ \ \ Q. CQ ...J \ \ \ + \ \ 0 il'l ~ S.lNnO:> -1l5aFigure 42 Integrated Reaction Cross Section for 3 6 Li (p, a)He (Page 30) The low energy data indicated by the solid line are those of Sawyer and Phillips, (Sawyer, 1953). The data indicated by solid dots are those of Marion, Weber and Mozer, (Marion, 1956). normalized to measurements made in this experiment. The energy El (Mev) is the energy of the proton in the laboratory system. The curve indicated by the open circles is that calculated for a 5/2- p-wave resonant state as shown on page 49. The curve indicated by crosses is the difference between this curve and the measured curve and is the s-wave reaction background assumed in the analysis of the scattering data. (See Table 5. ) -1 15 - I NTEGRATEO REACTION CROSS-SECTION -6 3 LI (p,a)He + + + 40 20 -1l6a- Figure 43 Amplitude Matrix Diagram (Page 35) The elements of the amplitude matrix (equation 68) are displayed for the case of s- and p-waves. For convenience the factors ib 1/2 i("1+"l,-2"lO) lC" (21 + 1) e are left out and only the Clebsch GO.rdan a coefficients, the exit spherical harmonics and the nuclear amplitudes are given. The general notation is The amplitudes within each box add and correspond to coherent processes. The cross section for scattering then is found by squaring each box. adding them and dividing by six. -116- AMPLITUDE MATRIX L;'(p,p)L;' "...:-.,. i .. 1.T +f!. y.+ ft u 'I'ltpp Y.', . ,, T , , ' , I:. 1 .T -$1:, •• 'C -jol!, •• Y,~ 'H~ •• Y,' .Jr f! t Y," ..r:; f!. Y,· -.rrI~ •• Y, ' -.Jj I:,,, Y,' •.jSI!, " Y,- -,' IItfpp Y.' , -f,I;,,, Y,' .~ -.J4 I:,,, Y,' -~ I!,,,y: +1 'J I!,,,Y,' , 1 -1 ..jfI!",Y ' pp "i : 1 liP VI:, .. Y: i-, -fol!,,, y' ·f.I!I" y' .. • , , 'IIIi pp y'• +1 -J I III"Y, ' VI:I .. Y: , II y' +;: 'IPP , .jtl~" Y,' - • I'Jlp(! y', f:. Y,-< pp -$ f!hp Y.' -$ f!IPP Y,' *T , .. •.g• , • ' I'Bpp Y.' -11: , Y, ' III" Y,' '~Ilt"y, ' IT pp f. I Y', - ifupp ft +f:." Y: 'IIUpp y'. -J\' II,,, Y,' ., .~ I!/pp y ' +B" , IUpp I y.', • IIHoP y.'' -JI III" y: -~I:I" Y: .~ I!,,, Y,' +-T , , I" ,-; .$111 .. y. -I -h111P. Y. -.jX 1:1 " Y,' ' IIUPI' y', -.)r I;, .. Y: {.kll - is upp Y'. .$ 1:1" 'c I ' 1"11/,, Y, +T tI~ , II y', , I' y', +ii Up p .,1 IIHpp y' -f,I:". Y: Y: -$1:,,, +~f!fPp Y, ' I , . i ,- . , -$1: 1" y. -.j;\ II,,, Y,' ..Jh t: hp Y," fc·f Uu Y. ' +,' IItiP!) y', -If:,,, Y: -1l7aFigure 44 Energy Level Diagram for Mirror Nuclei Li The corresponding nuclear levels of Li here with their assigned spins and parties. 7 7 and Be 7 (Page 39) and Be 7 are shown (Ajzenberg-Selove, 1959.) -117- 1.587 0.478 J=312 0.431 112- [-0.24J J~317 7 Be He3 +He4 -1l8a- Figure 45 Amplitude Matrix Diagram (Page 40) In constructing this diagram the assumption was made that only s- and p-waves contribute to the scattering and that the p-wave scattering is only through a resonant 5/2- state. (See figure 43. ) -118- AMPLITUDE MATRIX L;'(p.p) L;' "'""~, . . r, I i. - i , , i , i , . r T , . i .-' , , i ,-, i i t 'r , , , l ,r T r t:, H ~ + fl.l ~, y. • ft+f~uY: Vf: l " Y: .tt!,,, Y,. I • y". '~ .. fll-o +$ f!hP Y, I f,-f!l" Y: , fl y: +T Upp • +$ ftho Y,' ~f·Hpp' y" f, 'f: , .. Y: 'f' y. +'J Hpp , '~f:I" Y,' .~ f'Bop y'" t,.til .. Y. +;;0 . . -J t!." Y,. -1l9a- Figure 46 S-wave Scattering AlYlplitude DiagralYl (Page 44) The equation Y =A(8)X + B(8), where B(8) is the s-wave intercept, is plotted in the cOlYlplex plane for the six C. M. angles used in the experilYlent. At this low energy the six lines do cOlYle to an intersection within the experilYlental relative errors. The complex point (X. Y) must be the same for all angles at a given energy and lYlust fall within the inner circle with radius equal to (1 _ U)1/ 2 . §I U 6 (p p) Li 6 S- WAVE SCATTERING AMPLITUDE • ANALYSIS Ep(LAB). 0.892 MEV S- WAVE FIT W I ."_ c-:---.')o~ _ H a:c::::::=: _ .... \ I ' I ,... D " -120a- Figure 47 S-wave Scattering Amplitude Diagram (Page 44) The six lines at this energy do not intersect and indicate that the resonant structure at this energy cannot be described by s-waves alone. The dashed lines represent the addition of the H(e) intercepts to the B( e) intercepts. Since an intersection is still not possible, the s-wave. p-wave interference intercept ZG(e) must be computed for each angle. Figure 49 illustrates the solution which was finally taken as acceptable. ~ S-WAVE SCATTERING AMPLITUDE ..... ....... ' ANALYSIS 6 6 Li (p,p) LI 1!l9"7' 140"55' \ 1I<{48' 12(6' \ \ Ep(LAB)< I.B79 ME. V Y\~~ S-WAVE FIT \ 70.~''1 , \ \ \ \ \ \ \ \ \ \ \ , \ \ \ \ -...... -...... \ \ \ \ -...... \ \ \ \ \ \ \ \ '",,\ \ \R N o \ \ \ 1 \ \ \\ \\ '\ '\\ ' ,\ ~ '\ \ \ ~ \ \ \ \ \ \ -lZlaFigure 48 Scattering Amplitude Diagram (Page 49) This figure represents the region in the complex plane within the unit circle. 6 6 For the case of Li (p, p)Li the scattering cross section may be written in terms of three complex scattering amplitudes. The assumption is made that only s- and p-waves contribute and that the p-wave scattering is through a single resonant 5/ Z- state. The energy variations of the complex scattering amplitudes for this state (£5/Z' g5/Z)' for s-wave scattering through the 3/Z spin channel (f 3/ Z' g3/Z) and for s-wave scattering through the 1/z spin channel (f l / ' gl/ z) are indicated Z in the diagram. The energy variation of a fourth complex amplitude (X, Y) is also shown in the figure. The real and imaginary parts of this com- plex number are related to those of (f l / Z' gl/Z) and (f 3/ Z' g3/Z) in the following way: The curves are labeled as follows: s-wave X,Y amplitude. . • s-wave 1/z spin channel amplitude 0 s-wave 3/Z spin channel amplitude x p-wave 5/Z resonant amplitude A - 12 1- SCATTERING AMPliTUDE DIAGRAM ~ , ,, , ~ .0- _ I I , ,0--".. , ,, 1>-- - .1 . ' f- - ..' \ / -122a-Figure 49 S-wave Scatterimg Amplitude Diagram (Page 51) In this diagram the six lines C( e) is the Y =A(e)X t C(e) are plotted .where s- plus p-wave intercept. An explanation of the procedure used to obtain this final solution is given on pages 49-51. .t S-WAVE SCATTERING AMPLITUDE LI~p, p) LiS ANALYSIS Ep (LAB) c 1.879 MEV S-AND P- WAVE FIT (f.! -xl1 + (g,-y)1= z T .I- I I .... ,.I I ... I... 0I I I .,.. h:1 I",T'I\~\ I'j,.f'\' J I D fl l i (I- U)-(X ' .y l] [ N N -123aFigure 50 S-Wave - P-Wave Interference Cross Section (Page 51) The s-wave - p-wave interference term in the scattering cross section may be written as The coefficients IT sl(E l ) shown in this diagram are obtained from the analysis of the scattering data directly from the coefficient Z shown on page 51. as The energy El (Mev) is the energy of the proton in the laboratory system. -123- o ~~------~-----,------~-----,------,------,~ eo C\l C\l C\l W U ..-- Z W q 0::: C\l 0 W LL 0::: eo W ~ <D ~ I Cl... "': w > <! ~ C\l I (f) <D ~ --" W W IZ 0 > Q) I 0 If) 0 <;t 0 to 0 C\l (qW) IS..o 0 0 0- -124a- Figure 51 S-wave Phase Shifts (Page 35) The s-wave scattering amplitudes may be written in terms of phase shifts in the following way= I f + ig :; Ne 2i6 where (magnitude) I 26 =tan- 1 f (phase) I The quantities \ 6 / and 6 / are plotted here as functions of 3 2 1 2 laboratory proton energy and as C. M. proton energy. At low energy both spin channels appear to exhibit "potential" type behavior and would correspond to potential scattering from a charged hard sphere (equation 81) with a radius equal to about 2 fermis. The 3/2 spin channel exhibits this behavior over the whole region studied and in particular does not exhibit resonant behavior near 1 Mev. The 1/2 spin channel. however. does exhibit resonant behavior and would correspond to a resonant phase shift, 6 =tan which goes through 'Kj2 of r:::. 1. 44 Mev. -1 r 2(E R -El at about E (Lab):: 2. 76 Mev with a width p This is valid if r /r 1 for the resonance since = p then equation 79 becomes f + ig = e 2i( f + 6) = e 2i6 I -124- 90 r-----.-----,-----,---~---,...-----r_ -----.-, 80 S-WAVE SCATTERING PHASE SHIFTS 70 60 50 ~ SPIN CHANNEL-¥:) 40 o 30 20 10 Or-----B'-~--_r----_,~----+_----_.--+_~----~ -10 -20 -30 t SPIN CHANNEL+-¥ +