w v ,. ....-.a eus O O O O FJFHAPJ I O O uawra thbub O O O twa b 0 U1 .1}. UI H blbhb o o o \JOWU'I 5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.3 O wsbubuawcuunb61NbonH~h4H O 0 RNA NF‘ UiubUJNH mmmmmmmmmmmmmmm NH COLLECTIVE MODEL ANALYSIS. . . . . . DWBA Theory . . . . . . (p, p ') in Collective Model . Vibrational Collective Model. . Vibrational Model Parameters and Reduced Transition Probability Optical Model Analysis. . . . DWBA Calculations . . . . . Representative Results of Collective Model Fit . . . . General Results of Collective Model Analysis. . . . . . . . L-Transfer Assignments and Nuclear Deformations . . . . . . Reduced Transition Probabilities The 1‘ States. . . . . . . First Excited O+ State. . . . COMPARISONS WITH OTHER EXPERIMENTS. . . Energy Levels. . . . . . . Spin Identification. . . . . States Below 6.585 MeV. . States Between 6. 750 and T=1 Analog States . . . . States Between 7. 6 and 8. 8 MeV . 7.558 MeV. The High L- Transfer and Level Above 9 MeV. O O O O C O 0 Comparison of 6' s and G' s. . . MICROSCOPIC DESCRIPTION . . . . . . Theory . . . . Effective Interaction . Wave Functions . . . The Calculations. . . Transition Density . . Form Factors . . . . Results and Discussions The 1’, T=0 State . . The lst 3‘, T=0 State . The 2nd and 3rd 3‘, at omooooooooo T=0 St The 5‘, T=O, 1 States . . The Unnatural Parity States . Discussions on Even Parity Stat Systematics . . . . The lst Excited 0+ State . . S (D comet-0.0.0.00. V Page 54 54 57 58 60 62 64 68 72 73 79 85 85 92 92 94 96 97 101 102 103 104 109 110 113 117 122 122 123 123 127 127 131 134 134 140 140 144 Chapter VII. SUMMARY AND CONCLUSION. . . . REFERENCES . . . . . . . . . . APPENDICES . . . . . . . . . . I. Plotted Angular Distributions . II. Tabulated Angular Distributions. vi Page 146 149 156 157 195 Table II-l II-2 III-1 IV-1 IV-2 IV-3 IV-4 IV-5 VI-l VI-2 VI-3 LIST OF TABLES it Isot0pic and Spectrographic Analysis of Ca48 Target Used . . . . . . . Contributions to the Energy Resolution (40 MeV Proton on Ca40). . . . . . Energy Calibration and Determination for Ca40 (in MeV) . . . . . . . . . Optical Parameters . . . . . . . Collective Vibrational Parameters, Ca40 (p,p'), Ep=40 MeV. . . . . . . .4 Collective Vibrational Parameters, Ca40 (p,p'), Ep=35 MeV. o o o o o o o Collective Vibrational Parameters, Ca40 (PIP.) ' Ep=30 MeV. o o o o o a o Collective Vibrational Parameters, Ca4o (p,p'), Ep=25 MeV. . . . . . . . Spin and Parity Assignments of States in ca40 O O C O I O C C O O C 0 Comparison of Nuclear Deformations, 6L(F)o o o o o o o a o o o 0 Comparison of Reduced Transition Prob- abilities, G(sp) in Weisskopf Single Particle Units. . . . . Comparisons of Reduced Transition Prob- abilities Between (p,p') and (a,a'). . RPA Wave Functions Used (Given by T. T. K110) o o o o o o o o o o o 0 Transition Density Function . . . . A(1)(Ag) for the K-K Force . . . . vii Page 14 21 53 65 81 82 83 84 95 105 107 108 124 125 126 Table Page (1) 2 VI-4 A (10) for the K-B Force . . . . . . 126 VI-S Ratio of Total Cross Sections o[D+E]/0[D] . 131 viii Figure 2.1 LIST OF FIGURES Cyclotron and beam transport layout used in this experiment at the Michigan State University Cyclotron Laboratory. . . . . . Target chamber and detector arrangement . . . Block diagram of the electronics used with the Ge(Li) detectors. . . . . . . . . . . 40Ca(p,p')40Ca* spectrum taken at eLAB=3l‘20 for Ep=34.78 MeV. Linear scale in count number is used for illustration of the relative strengths of various peaks at this angle. . . 40Ca(p,p')4OCa* spectrum taken at GLAB=31.7° for Ep=39.83 MeV. . . . . . . . . . . 40Ca(p,p')40Ca* spectrum taken at GLAB=31.2 for Ep=34.78 MeV O O O O O O C C O O 40Ca(p,p')40Ca* spectrum taken at GLAB=31.79 for Ep=30.04 MeV . . . . . . . . . . 40Ca(p,p')4OCa* spectrum taken at eLab=31.7° for Ep=24.93 MeV . . . . . . . . . . Cross sections of elastic scattering of proton from Ca4O at 25,30,35 and 40 MeV. Curves between data points are drawn to guide the eyes and have no physical significance . . . . . The decomposition of doublet at EX=8.558 MeV . The decomposition of doublet at EX=7.539 MeV . The decomposition of doublet at EX=8.097 MeV . ix Page 16 22 23 24 25 26 38 ‘44 45 46 Spectrum fits using two superimposed standard peaks for the analysis of 6.905, 6.926, 6.944 triplet . . . . . . . . Optical model fits to the experimental elastic scattering results at Ep=25 to 40 MeV . Summary of the DWBA calculations using collective model F.F. for 1:2 to i=8 at Ep=25 to 40 MeV . . . . . . . . . . Typical results of experimental distributions and collective model fits (solid curves). The upper parts show the variation of shape of the experimental distribution with respect to beam energy . . . . . . . . . . . Experimental distributions of i=2 states and collective model fits (solid curves). The dash curves are those of 3.900 MeV state. . Experimental distribution of i=3 states and collective model fits (solid curves). The dash curves are those of 3.732 MeV state. . Experimental distributions of i=4 states. Solid curves are collective model fits. Dash curves are those of 6.502 state. . . . . Experimental distribution of i=5 states and collective model fits (solid curves). The dash curves are those of 4.487 MeV state. . Experimental distributions of £=6 and i=7 states, and collective model fits (solid cur-V85) O C O O O C O C O O O O 0 Experimental distributions of i=1 states. The solid curves show the poor collective model predictions . . . . . . . . . Results of generalized collective model calculations based on Satchler's theory . . Experimental distribution of the first 0+ state and empirical F.F. fits . . . . . Page 49 63 67 69 74 75 76 77 78 86 89 90 Energy levels of experiments . 40 Theoretical and experimental energy negative parity states of Microscopic DW state . . . Microsc0pic DW 5:0 state . . Microscopic DW T=O state . . Microscopic DW T=O state . . Microscopic DW T=O,l states . Microscopic DW T=O; lst 3‘, states . . . Systematics of the even-parity states in Ca observed in this experiment. T=1; calculations calculations calculations calculations calculations calculations lst 4', xi T=0,1; 40Ca for the for the for the for the for the for the 6’ .1evels for lst 3rd Ca observed by various , T=0,1 40 Open circles are those from other experiments . Page 93 119 128 129 132 135 136 138 141 CHAPTER I INTRODUCTION 40Ca is a nucleus of considerable theoretical interest because of its double closed shell structure. The degree of deviation from this simple structure is of great interest. Recent advances in the theories of nuclear shell models (RPA and deformed), effective nucleon—nucleus force, and the distorted wave treatment of direct reaction enable one to formulate a microscopic description of the inelastic scatter- ing of protons by nuclei. A microscopic DWBA theory includ- ing anti-symmetrization for the (p,p') reaction at medium energy has been developed at Michigan State University and elsewhere. 40Ca is one of the nuclei of interest. (However, 40Ca in the previous inelastic proton scattering data for range of 20-50 MeV were insufficient to provide a test for this theory. Rectification of this situation is one of the main motivations of performing the present experiment. The 40Ca nucleus was chosen because in order to test the (p,p') reaction as a probe of nuclear structure, one needs: 1) a target which allows to examine all the com- ponents of the proton-nucleus force. 2) a target in which the eigenvectors describing the excited states are well established both experimentally and theoretically. 3) a target for which good optical model parameters exist. The structure of 40Ca has also been investigated in other experiments such as (a,a'), (e,e'), (3He,d) and (d,n). The (a,a') reaction is a predominantly surface dominated reaction and it leads to diffraction scattering. It provides information for L-transfer for the excited normal parity states, as well as the information on the isoscalar com- ponent of the projectile—nucleus force. The (e,e') reaction gives reduced electromagnetic transition probabilities and multipolarities. The (3He,d) and (d,n) proton stripping reactions allow one to study a component of the vectors of the excited states. Previous 40Ca(p,p') experiments giving some angular distributions were reported by Gray eE_gl. (Colorado) and Yagi §E_§1. (Japan). The experiment at Colorado was per- formed at 14 and 17 MeV with resolution about 80-100 KeV. The one at Japan was done at 55 MeV with 500 KeV resolution. The present experiment was conducted at 24.93, 30.04, 34.78 and 39.83 MeV beam energies. The target used is 99.97% enriched and is 2 mg/cm2 thick. Spectra were taken simul- taneously by two surface barrier Ge(Li) detectors which were fabricated by the author of the present work and his collaborator. The overall resolution (FWHM) was 30-35 Kev. A goniometer was specially designed to facilitate the use of Ge(Li) counters and to provide a mechaniSm for trans- ferring Ca targets into the scattering chamber under vacuum environment. Angular distributions from 13° to 970 for elastic scattering and about 40 inelastic states were obtained. The weak excited states were of interest and the develop- ment of the high resolution Ge(Li) detectors with the best peak to valley ratio obtainable was directed toward this goal. The usefulness of thick and enriched target is also apparent. In this thesis, the experimental apparatus and methods of obtaining and analyzing the data are described in Chapter II and III. The collective model analysis and the extraction of nuclear deformation are presented in Chapter IV. Chapter V is devoted to the summary of results of experimental sources. The microscopic DW calculations are described in Chapter VI, where the effective force and RPA wave functions used in the calculations are discussed. CHAPTER II EXPERIMENTAL APPARATUS AND PROCEDURES 2.1 Cyclotron and Beam Transport System 2.1.1 The Cyclotron The proton beams of variable energy were produced by the sector-focus cyclotron (BI 61) at Michigan State University. The principle and the details of the design of the machine as well as its operation have been reported elsewhere (B1 66, Go 68). The most important objective in the operation of the cyclotron is a well tuned beam with high extraction efficiency. This can be accomplished by setting the main magnetic field precisely, centering the beam carefully to reduce the effect of RF ripple and select- ing a narrow phase group to get an optimum single turn (resonant) extraction. The H+ beam is extracted at a radius of about 29 inches (212 turns), using a small first harmonic bump field to induce a coherent radial oscillation, together With a guiding electrostatic deflector and a focusing air-core magnetic channel. The beam is then balanced on the exit slits 81 as shown in Fig. 2.1. Typical internal beam currents were 1 to 5 microamperes and extraction efficiencies were about 80% during this experiment. 2.1.2 Beam Transport System The external beam transport system is shown in Fig. 2.1. Detailed discussions of the optical properties of the beam and of the energy analysis system have been published (Ma 67, Sn 67, Be 68). M1 and M2 are horizontal bending magnets used to align the beam through the object slit S3 and the divergence slit S4. 32 is a vertical slit which was not used in this experiment. Two quadrupole doublets Q1' Q2 and Q3, Q4 are used to focus the beam on S3. The distance between S3 and S4 is approximately 48 inches. Thus the openings of 83 and S4 determine the divergence of the beam. M3 and M4 are two 45° analyzing magnets the fields of which are adjusted so as to direct the beam to the image slit, SS. Nuclear magnetic resonance fluxmeters (Scanditronix, NMR-656C) in M3 and M4 are used to measure the magnetic fields which determine the energy of the proton beam. QS and 06 are quadrupole magnets used for refocusing. The beam has to be balanced on S3, S4 and SS simultaneously. The balancing can be exercised by adjusting the current on each side of the individual slit. The geo- metry of beam can be viewed from the scintillators in front of S3 and SS. After these conditions are satisfied, beam is then deflected into the target chamber by the distributing \\ N \\\\\\\\\\N E CYCLOTRON \ I - \\\\\\\\\ V/AW/A; “1:: Figure 2.1.--Cyclotron and beam transport layout used in this experiment at the Michigan State University Cyclotron Laboratory. magnet M5. Two more quadrupole doublets Q7, Q8 and Q9, 010 are used to focus the beam on the target. For the final beam preparation the following pro- cedures were exercised. A plastic scintillator with a 1/16 inch hole in the middle was used for centering the beam on the target. The sharp and greatly enlarged image of the hole and the boundaries of the scintillator viewed by TV camera were first marked on the TV screen. When the beam hit the scintillator, the location of the spot could be seen clearly and the best focusing and centering could be achieved. In addition there were two more devices used for maintaining the correct alignment of the beam in the course of the experiment. One was the neutron background in the vicinity of the target chamber, the other was the current monitored by a tantalum ring which is shown in Fig. 2.2. The neutron background and the ring current must be kept in a minimum with respect to the beam current detected at the Faraday cup. The ring current was probably due to the particles which were scattered in the slit, SS. 2.1.3 Beam Energies In this experiment, typical slit apertures were about 25 mils for 83 and SS, 100 mils for 84. These settings yield a beam divergence of 10.8 milliradians which is equivalent to an 8-10 KeV energy spread on target at Ep=40 MeV. The absolute energies of the proton beams were cal- culated from the NMR reading in M3 and M4. The uncer- tainty in absolute scale was 0.1% (Ma 67). .The calibrated absolute energies for this experiment were 24.926 MeV 1 25 KeV, 30.044 MeV t 30 Kev, 34.775 MeV i 35 KeV and 39.828 MeV i 40 Kev. 2.2 Target Chamber The goniometer used in this experiment was designed by K. Thompson (Th 69). The target chamber, designed by C. Maggiore (Ma 70), is 16 inches in diameter and shown in Fig. 2.2. Two beam pipe adapters were plugged into the chamber with a double O-ring seal. On the right hand side (following the beam direction), an opening of 100° wide and 1% inches high was covered by a 5 mil stainless steel slid- ing seal. A block of brass with two 3/4 inch brass tubes was soldered to the steel sheet. The center of the chamber could be viewed through the brass tubes. This block was attached to the main arm so that the sliding seal could be moved in either direction by the action of the arm. There were baffles made of 50 mil tantalum sheetwwhich encircled the target holder, two standing on the bottom, and another two hanging from the top of the chamber. The vertical opening of the strips was 1/2 inch. The end of the brass tubes was covered by tantalum rings with 3/8 inch holes. This arrangement was designed to minimize the multi-scattering into the detector. .usmammcmuum Houooump new Hmnfimgo pomumell.~.m musmflm /, 34mm 62.9.5 . ....wo Juwhm mmezzhw N .kwo 0 \ wank 03.5300 . - . 24mm 24mm - - . "5.5404 mm X. 24mm «323$ \ 5.538 mo: >h 202 3002.3 20.7.23. 10 The detectors were coupled to the tubes bY sliding O-ring seals. Detector 2 was always placed at the smaller angle tube so that the solid angle was constant from one energy run to another. The angular separation between these two detectors was also mechanically fixed and was measured to be l4.7° (see Section 3.1). The mylar window on the detector cap was the only material through which the scattered protons had to pass before being detected. On the other side of the chamber, an Opening covered by 1/2 mil kapton foil served as viewing window for the TV camera in the monitoring of the beam spot. It also allowed scattered particles to be detected by various kinds of monitor counters. A secondary arm provided a convenient platform for mounting these counters. 'The target holder could be rotated and moved vertically by remote control. On the top of the chamber was the target transfer system (Th 69) and the coupling pipe to the diffusion pump. The vacuum inside the chamber was maintained at about 5x10"3 microns and monitored in data room by television. 2.3 Faraday Cup and Integrated Charge The monitor counter becomes standard equipment for normalization in this thesis. The Faraday Cup used was a half-inch aluminum beam stop isolated from the target chamber and shielded by concrete blocks 6 foot wide and 7 foot high. As seen in Fig. 2.1, additional shielding was 11 provided by a cylinder of paraffin surrounding the beam pipe about 3 feet from the target chamber. The neutron background was reduced about 10 times below the case of no shielding. Data so taken were much cleaner and the lifetime of detectors were extended. The relative integrated charged was measured by an Ortec 439 current digitizer along with an Ortec 430 scaler. The current digitizer triggered the scaler every time after it has collected preset charge level (in the 12 to 10_8 coulomb). From the calculation of order of 10- absolute cross section, the charge lost were found to be ~30%. There were cases in which the charge was fully collected. Those cases were found about 30% higher than those after loss. The causes of charge loss were probably due partly to multiple scattering after the beam travelled through the target, to leakage to the ground and to the malfunction of the current meters. 2.4 Detectors Two Ge(Li) surface barrier detectors were used to take data simultaneously. These detectors were fabricated in this laboratory, one by the author of this thesis and 48ca(PIP') the other by C. Maggiore who investigated the reaction using the identical experimental setup. Details about the fabrication of Ge(Li) detectors are discussed in Appendix I of Maggiore's thesis. 12 As shown in Fig. 2.2, these two detectors were fastened to the coupling mechanism on the sliding seal. Detector l and 2 were always attached to the same coupling tube and the distances from the detectors to the center of the target chamber were also fixed. The angular separation was measured to be l4.7° (see Section 3.1). The monitor counter employed throughout this experi- ment was a Ge(Li) detector of side entry geometry mounted in a Harshaw satelite cryostat. It was mounted outside of the target chamber on a secondary arm whose angle CODld be manually adjusted. Scattered protons were detected after passing through the 1/2 mil kapton windows of the target chamber, about 1/2 inch of air and then the 1/4 mil diminized mylar window of the detector cap. The overall resolution of this counter obtained with the above arrange- ment was about 100 KeV. The peak to valley ratio Was 1000:1. The electroniCSIHMaiwill be described in Section 2.6. 2.5 Target The target used in this work was a 99.973% isotopically 40 foil. Its thickness was 2 mg/cmz. This tar- enriched Ca get was purchased from Oak Ridge National Laboratory (ORNL) and shipped in a vacuum tube. Mounting the foil on a target frame was done in Argon atmosphere. The mounted tar- get was immediately placed in a target storage chamber (Ma 70) 13 which was evacuated to a vacuum of the order of 5x10_6 mm by an absorption pumping system. After having been transferred from the vacuum ship- ment tube into the storage chamber, the_target was never exposed to air or argon. This was accomplished by the coupling scheme of target storage and transfer system designed by K. Thompson (Th 69) and C. Maggiore (Ma 70). The target thickness of 2 mg/cm2 was so chosen that the "signal to noise" ratio would be good enough to observed the first excited O+ state and that higher efficiency of data taking could be achieved, without unduly high beam currents on target. The energy straggling of protons passing through this target at normal incidence was about 12 KeV more or less. It increased to 18 KeV when the target plane was set at about 50° with respect to the beam. ‘The amount of contamination in the target due to oxidation and condensation of pump oil molecules were obtained from the elastic scattering data. It was found that the thickness of oxygen was about 0.01910.002 mg/cmz, carbon 0.002610.0003 mg/cm2 and hydrogen 0.0017i0.0002 mg/cmz. There was also a small amount of F19 whose elastic peak showed up clearly in some spectra. However, there is no 14 proton elastic scattering data in the energy range of 20- 40 MeV, and therefore the amount of F by ORNL is listed in Table II-l. * TABLE II-1.--Isotopic and Spectrographic Analysis of Ca Target Used. 19 was not estimated. The isotopic and Spectrographic analysis supplied 48 Isotopic Analysis Spectrographic Analysis Ca4O Ca42 Ca43 Ca44 Ca46 Ca48 99.973% 0.008 0.001 0.018 <0.001 0.001 A9 A1 B Ba Co Cr Cu Fe K Li Mg Mn <0.02% <0.05 <0.01 <0.02 <0.05 <0.05 <0.05 <0.02 <0.01 <0.01 <0.05 <0.02 MO Na .Ni Pb Pt Rb Si sn Sr Ti V Zr <0.05% 0.01 <0.05 <0.05 '<0.05 <0.02 <0.05 <0.05 0.02 <0.02 <0.02 <0.1 * supplied by Oak Ridge National Laboratory 15 2.6 Electronics Fig. 2.3 shows the block diagram of the elec- tronics used in this experiment. The electronics used for detector 1 and 2 are identical and only slightly different for the monitor counter. The 1500 volt bias supply (Model 250) for the data taking detectors was purchased from Mech-tronic Nuclear Corporation. The voltage applied to detector 1 and 2 were 1500 and 1200 volts respectively. Modified Ortec 109A preamplifiers were used for the first stage amplification. The modified model was designed for up to 90 MeV proton detection using Ge(Li) counter (W1 67). A shaping amplifier board was added between the charge sensitive loop and the cable driver of model 109A. The pole-zero network and voltage amplifier were bypassed. The shaping time constant T was 2p sec for both differentiator and integrator. The preamplified pulses were fed into the second stages of Tennelec TC 200 amplifiers. This section of the amplifierswas found to have the least noise at the time when this experiment was being performed. Since the shaping pre- amps were used, only one step of differentiation and inte- gration in the TC 200 amplifier was needed. Therefore, this arrangement of preamplifier and main amplifier was capable of providing optimum electronic resolution. The outputsignals from the TC 200 amplifiers were always monitored by a RM41A Oscillosc0pe (Tektronix, 16 .muouoouoc Aflqvmo may nuflz poms newcouuomam may no EBHmMHU xooamul.m.~ musmflm 41¢ £24 , om. oz 23 - we”. \ , «umhfir 7343 n -555- - _ 1:4 — kw _ _ 34 L 08 o» mh mo msumsmuum m>flumamu may mo coaUMHUmsHHw Mom pmms ma umnass assoc ca mamom Hmmcwq m nmq .>mz m~.qmu u now .N.Hmu a an :wxmu gunpowmm «moovi.m.mvmu u-.v.~ musmem ov mmmsSz ...wzzSio coon 8mm ooom com .1 08V com ‘ 14.; w 1 1; 1.1.14 ...” .,_._._:_._ m m m a mm_m 1 u e m m 1 ( .1 _ _. _ f _ ktxxxtrfrffF/ffr E we m... N .m- ....m . wmw m iminwmmmmwmwwumwflimmmw >m2mh. NM... aw ( L ¢ A.n_ .n_vonv O¢ o o If) WENNVHO / SanOC) 3'0 LO)OG€€ (O rum,l LE)Z€LY “2)0065 € ass Lmius uqus iwme9 s on; tv)csvz (.912) 629‘). 29%1 tvimu Lamlam gene? wv LIzams acne )iwm -9 9206 .£ Ibr6 = 39.83Mev “WSW. 400m P. P’) 59 J— k 11 1" 4f- W 317° (P'd) eLab I J ISOO CHANNEL NUMBER N O O jauNVHS/smnoii > (D 2} m 00 O‘ M II 04 fifl H O LH o l‘ H m (XX) Figure 2.5.-40Ca(P:P')40C3* spectrum taken at eLab 24 .>0: ah.vmumm mo“ o~.Hmunon um sexed asuuommm no ..m.mvuo e cc Il.m.~ mnsmflm ow ”1829.425 com. axXH cow .1; . ( ”'17) D..- (269)O.. -— 11s a" (p‘d) 1‘60. :0 \ExE 7r fr frfffrr _ ..m .m- 23¢ 91.20 I — . Sci... 19 GSSBSBBBSLLLLLBBBBBGGSI IQ I u we meeeeeéeéi-em, 5&3 en. m 1.0. We... Help”... m a E. 588 . b L 'IBNNVHO/SINOOO (‘l'bln'l ‘ . ( 5 I5):)“ _. 3000 (.Z 0929 (MW) 3,. .— tv 2099 E 1.1.99 1 ('9 ”‘9 \ ‘§ 9069 q -l ‘ 3- ) ”7?? _/ (269:0..— “—8 8 I500 CHANEL NUMBER Lida @0315 9 ' ' cue- (.Z ”IVS 319 8998 “efe243 )3 Hfi6__ -9 9986 (5)990!”— :> a) 2 0 V N f‘c) __ 0- . .O ’0 II (in a v n O 8 mm (13’ 0 ¢ l I 1 E3 73 W5 73 IBNNVH0281NDO0- =30.04 MeV. P . 31.7° for E * Figure 2.7.-40Ca(p,p')4oCa spectrum taken at eLab 26 ('5 Mo" (.0) 099? —- (-9) 291.9 -—- (.2) 0069 f (.9) 1.859 —- (z)mas 6v)ozzs Lvmeum (J)M£9 I" N In" if): .§Q¢:$n¢3 mmmmmmm U I jfififi‘h ml ( WIT/n” F [s N O =24.93MeV 3| 7° eLab 40Co( P.P') Esp — (DP) 1 l 04 l I: :IEINNVHZI/SanOD N .— O O 3000 2500 1500 CHANNEL NUMBE R I000 500 31.7° for 35:24.93 MeV. . * Figure 2.8.-40Ca(p,p')40Ca spectrum taken at eLab CHAPTER III EXPERIMENTAL DATA ANALYSIS 3.1 Laboratory Angle Calibration The laboratory angle for each spectrum was deter- mined by the energy separations between the elastic peaks of Ca40, O16 and C12 and the 3- excited state of 40Ca at 3.731 MeV. Using the program FASTKINE written by W. Plauger, the relativistic kinematics of the scattered protons were calculated for each nucleus in 0.1° steps in the vicinity of the estimated laboratory angles. The laboratory energies for each nucleus were plotted with respect to laboratory angle together on one linear graph. Thus the calculated energy spacings could be read continuously as the function of laboratory angle. The experimental energies of the forementioned peaks were calculated from the positions of their centroids. With the known energy difference between the 40Ca [0.000 MeV] and the 40Ca* [3.731 MeV] states at a particular angle, ‘the energy spacing between these four peaks were computed. Iiowever, without knowing the exact angle, the energy calculation is only approximate. It was, therefore, Inecessary to reiterate this angle and energy calibration 27 28 procedure. Since the energy difference between the 40Ca* 40Ca [0.000 MeV] states changes slowly with [3.731 MeV] and respect to angle (about 0.8 KeV/deg. at 25° and 1.7 Kev/deg. at 100°), most computations required only two iterations. By fitting the experimental energy differences to those calculated, the laboratory angle was determined. For laboratory angles less than 28°, the H(p,p)H reaction was also used. The fact that the kinematics of this reaction is stongly dependent on angle provided an acute test of the accuracy of the method described above. The agreement between these two calibration methods agreed within 0.04 degree. The effect upon the accuracy of determinations of the laboratory angle of the uncertainties in the beam energy and in the centroids of peaks was studied. Two kinematic cal- culations were done using Ep=35.000 MeV and 34.775 MeV. The laboratory angles calibrated by these two calculations were within 0.1 degree. When the centroids were allowed to flux- uate f0.2%, the calibrated angles varied by 10.04 degree. In addition to the above, other checks of the angle calibration were made. For example, the angular separation between two detectors used was mechanically fixed. This separation was obtained by computing the difference between the calibrated angles of these two detectors when they were taking data simultaneously. The angular difference between the counters was found l4.7° throughout. On the other hand, 29 these two detectors overlapped at about 27° where the differential cross-section of the elastic peak changes drastically. Should the angle be measured incorrect by more than 0.1 degree the matching of the elastic angular distribution would be very difficult. In this experiment, each distribution was matched to with 1%. 3.2 Normalization of Data 3.2.1 Dead Time Correction Dead time corrections were made for all spectra including those taken by the monitor counter. The per- centage corrections were obtained by taking the ratio of counts registered by the scaler to those registered by the zero channel of the analyzer (see Section 2.6 for elec- tronic setup). The dead times for most spectra were under 2%. For only a very few cases (5 out of 100) in which the detector was set at small angle, were corrections found to exceed 5%, the largest being 12%. 3.2.2 The Monitor Counts The entire monitor spectrum was taken by the ND 160 analyzer for each run. In the early stage of data analysis, the effect of the window width of the differential dis- criminator was investigated, for it was feared that elastic counts might get lost in the long tail of background. As mentioned in Section 2.4, the peak to valley ratio of the monitor counter 30 used was about 1000:1. Consequently when a window was con- sistently chosen, the relationships between monitor counts, integrated charges and target angles were found to remain almost the same. The monitor counts used for normalization were obtained by setting a window which covered all the 4O 16 elastic peaks of Ca , O and C12 so as to minimize tail losses even though they were small. 3.2.3 Charge and Target Angle Ratios of monitor counts (after dead time correction) to integrated charge were computed to examine the charge collection system for relative errors. An average value was obtained for each target angle and deviations from the average were also computed. Most of these deviations were less than 1%. The average value of ratios also provided a way to check the target angle. The ratio of two mean values should be equal to the ratios the cosines of the correspond- ing angles. When the backlash of the target frame driving system was treated properly, the readout for target angle was found accurate to :1 degrees. The consistency between monitor counts and integrated charge enabled either of them to be used for normalization. In this experiment monitor counts were preferred because they were obtained by a somewhat more reliable and con- trollable electronic setup and hence believed to be more accurate. 31 Throughout about 100 data taking runs, there were only two successive runs in which the integrated charges showed a 30% discrepancy. On the other hand, from the calculation of absolute normalization, the integrated charge was found consistently 30% lower than expected. Probably this was due to a loss of charge between the Faraday cup and current digitizer system. 3.2.4 The Solid Angles of Detectors The solid angles of the two detectors used in this thesis and their measurements in area and distance from the center of target chamber are listed as follows: Detector l Detector 2 Width 2.2 :0.05 mm 1.9 i0-05 mm2 Area 14.5 10.4 mm2 10.5 i0-35 mm Distance 32.45:o.25 CM 36.65 10-25 CM 4 Solid Angle 1.38:0.04 x10‘4Sr 0.786i0-024x10 Sr The relative ratio of solid angle of detector 1 and 2 so obtained was AQl/AQZ = l.75510.105. The ratio of effective detection efficiencies was determined by matching the relative differential cross section for the elastic peak at the overlap angle of 72°. This was done for each beam energy. The result of the four measurements yield an average value of 1.7810.01, for the efficiency ratio. 32 3.3 Method of Normalization The differential cross section is defined as the probability of finding scattered particles through a unit solid angle per unit incoming flux per unit scattering center. It can be written as 99(0 )= Nevent d0 Lab NSCatt-I-Afloeff where Nevent is the number of events detected within solid angle A0 N is the number of scatterers per unit area scatt I is the number of incoming particles eff is the efficiency of the detection. (equal to 1 for ideal detector). For the present experiment, these physical quantities were more specifically defined. Nevent is the number of counts extracted from a spectrum after correction for analyzer dead time loss. Nscatt can be obtained by calculating the number of atomic weight per unit area, that is t/A where 2 t is the thickness of target in mg/cm and A is the atomic weight in mg, and then converting it to the number of target 23). Normalization to target angle nuclei (t/A x 6.023 x 10 should also be taken into account. The number of incoming particles I is computed from the recorded integrator charge. I is also prOportional to the monitor counts in a given run. This ratio may differ from runs with unequal target angles. 33 For the convenience in the data analysis two simpli- fied expressions for differential cross section were used do 55(0Lab)=Counts x Abs. Norm. factor =Counts x k'% where k is the relative normalization factor belong to a given spectrum. The former one implies that once the peak counts are given, the cross section can be obtained by just one step multiplication. The latter is used for computing the amount of contaminants in a target. To test the overall accuracy of the calibration works described in Section 3.1 and 3.2, and to examine the correctness of the formula used in computing absolute differential cross sections, several calculations were tried. For example, the hydrogen peak counts were first extracted from a mylar spectrum at 6 =26.7 degree and at Lab Ep=40 MeV. By assuming the efficiency of the detector be 98.75% (Ja 66), the absolute cross section in the center of mass system was found to be 14.6 mb/sr (0CM=53.96°). This was about 30% higher than ll.1210.5 mb/Sr obtained by Johnston (Jo 58). Calculations for C12, 016 and Ca40 and comparisons with other experiments are listed below where 9CM is the angular location of a maximum or a flat region in the distribution. 34 do do . Target Ep 9cm 351A 351A Ratio (Trial Cal.) (Ref., Abs.ERR.) .(Ca1./Ref.) c12 40 60° 13.9:2.4% 10.3:2.0% 1.3516.l% (B1 66a,15.0%) 16 0 o 40 50 26.212.0% 20.2:1.0% 1.29:2.8% (Ca 67,il.7%) Ca40 40 41° 126.010.2% 96.712.0% 1.31:5.4% (Bl 66a,15.0%) Ca40 30 46° 143.610.2% 110.1:1.7% 1.31:3.4% (Ri 64,13.0%) It can be seen that all results of the trial cal- culations were consistently 30% higher than other measure- ments indicating that the combined systematic error of the integrated charge and the detector solid angle was about 30%. We have attributed this discrepency to a malfunction in the integrator. Viewing this matter from another angle, one finds that once the charge loss of this experiment was corrected, good agreement between this experiment and various others was obtained. Most important of all, the results of elastic scattering from Ca4O obtained by ORNL and Oxford groups were confirmed. Consequently the elastic and inelastic scattering data of this work were believed to be normalized within 13%. 35 3.4 Treatment of Contaminant Data The main contaminants observed were H, C12 and 016. The hydrogen and carbon came from the deposition of pumping oil on the target while the oxygen came from the oxidation of the Ca during the mounting of the target foil. A complete analysis was made for C12 and 016. First, it was necessary to know the number of counts for the individual inelastic peaks of these two contaminants in a spectrum of interest. To do this, a mylar target was used to measure the ratio of counts of the inelastic to the elastic peaks at the identical angles at which Ca40 data were taken. This method provided a reference to monitor the intensity of the contaminant peaks in Ca40 spectrum, because the mylar spectra did not need to be analyzed in detail. Once the ratio of counts in the mylar run was com- puted, the number of counts for the same inelastic contam- 40 spectrum was easily determined as long inant peak in Ca as the elastic counts were known. The corrections for contaminants at small angles, where the C12 and O16 elastic peaks could not be separated from that of Ca40, required the knowledge of the thickness of each contaminant. To determine the amount of 016, a complete analysis was done as follows. Take the case of 40 MeV for example. The angular distribution of relative cross sections in laboratory system for the O16 elastic peak 36 was first obtained. This result was compared with the mea- surement reported by Cameron (Ca 67). Good agreement in the shape of the distribution was noted. This suggested 16 that the buildup of O on the target remained essentially constant in the course of the whole experiment. Secondly, 16 in the target was calculated by using the amount of 0 Cameron's data and the equations described in Section 3.3. Several values were computed over a few angles around 6 =50° where the distribution is flat. The average value 16 40 Lab of the amount of O in the Ca target used was found 0.019210.002 mg/cmz. Thirdly, the amount of correction for contamination in the number of counts in the composite elastic peak at 12° and 17° were obtained by inverting the procedure of the second step. data from ORNL was used (Bl 66a). The thickness of C12 was measured to be 0.00258:0.0003 mg/cmz, Similar correc- tions were made for 35, 30 and 25 MeV data. 3.5 Elastic Angular Distribution The elastic peak counts were obtained by first drawing consistent peak tails extended to each side of the peak and then calculating the area under the boundaries. Since the average peak to valley ratio was 5000:l, the uncertainty due to the extraction process was very small. Peak counts were then corrected for dead time loss and normalized by monitor counts to obtain relative cross 37 sections prior to the relative normalization between two counters. Relative cross sections for each counter were plotted and carefully matched at the overlap angle at about 72 degree and the accuracies of matching were checked at 27° (see Section 3.1 and 3.2.4). An average value of 1.78 for relative counter normalization was obtained. Although various measurements in this work would hypothetically enable us to obtain independent absolute cross sections, we have not done so because of the apparent large amount of integrated charge loss previously mentioned. Rather, our cross section normalization were obtained by normalizing our relative cross sections to the existing data reported in literature. For 40 MeV, data from Oak Ridge National Laboratory was used (Bl 66a). For 30 MeV, those from Harwell, England (Ri 64) was compared. It was found that the normalization factor computed from the comparisons at 40 MeV and 30 MeV agreed to better than 0.3%. There were no existing data to compare with for 25 MeV and 35 MeV. However, judging from the good agreement at 40 and 30 MeV, It was decided that the same normalization factor be used. The angular distributions of the differential cross sections for elastic scattering in the center of mass system are shown in Fig. 3.1. Data are tabulated in Appendix II. IO b C) (3 c: A4 13 13 v 11111 is r- . 4 (I 3 '0? ‘2 r- . ‘1 . g : t . )- II ‘2 1‘. IOZF ° 1 m b .1 ‘~ : 1 E : - <3 1 .. 5 \ b’ )CI \\:§”'zl ‘\ 1: ‘ “ZSMeV L lLlllll T TjVUU'I / ./ on an E < \. IO°:- ° 40 MeV —.-. l 1 L L l L L l 20 40 60 80 '00 '20 ecu (deg) MSU cvc Figure 3.l.--Cross sections of elastic scattering of proton from Ca40 at 25,30,35 and 40 MeV. Curves between data points are drawn to guide the eyes and have no physical significance. 39 3.6 Inelastic Angular Distributions Priot to the analysis of inelastic angular distri- butions, the spectra were subjected to careful inspection and study. Peaks which lie below 7 MeV excitation in the spectra were well separated and well resolved except for 5.24 and 6.92 triplets. These separated peaks were easily identified and were analyzed first. The region between 7 and 9 MeV was densely populated. A spectrum taken by Grace and Poletti with a magnetic spectrograph was used to help identify these closely spaced states. The absolute laboratory energies were calculated for 12 16 the inelastic states of C and O at the calibrated angles using program FASTKINE. The states involved were 012 0.000, 4.440 a“; /.660 MeV (Le 68) 016 0.000, 6.052, 6.131, 6.916, 7.115 and 8.890 MeV (Le 68) The kinematically determined energies of these states were tabulated and then transformed into channel numbers. Con- sequently the positions of these contaminant peaks were marked in each spectrum. The overall quality of all spectra were summarized in a chart which showed the conditions of each peak such as resolvability, intensity, freedom from contaminant, etc. After this preliminary inspection the spectra were ready for cross section analysis. The areas, centroids 40 and statistical uncertainties of the peaks of interest were computed by the program PEAKSTRIP written by R. Paddock. The output of this program was in turn used as a part of input of another program, RELTOMON also by R. Paddock which calculated the absolute cross sections for both the laboratory and the center of mass system. The output of RELTOMON included printed listings, graphs of angular distributions in usual 4-cycle semi-log plot and punched card decks. The results of the complete analysis will be dis- cussed in detail in the following sections and chapters. 3.7 Errors Aside from the statistical and normalization errors, the sources of other errors can originate from the uncertainties in background substraction, setting of peak boundaries and contaminant counts in the analysis. Most of the Spectra displayed a clean background below 7 MeV due to the excellent peak to valley ratio of the Ge(Li) detectors used. Above 7 MeV, the background is higher because of the greater density of states and the slow-dropping tail of the scattering from the degrader slit. The effect of uncertainty in background level assignment was studied by setting upper and lower limits for background levels to see the differences in peak counts. It was less than 1% for 3.731, 3.900, 4.487 MeV states and about 3 to 10% for others. 41 The effect due to errors in setting peak boundaries would be sizable for closely spaced peaks. Usually consistent boundaries were assigned before peak areas were extracted. A large amount of error may result when a weak Ca40 peak was overlapped by a strong contaminant peak, for example, the 6.131 MeV state of 016. If the net counts of the Ca40 peak were of the order of the statistical error of the contaminant peak, this datum point would be discarded. Extraction of the peak areas for weak states at small angles, typically 12° and 17°, was most difficult. This situation was characterized by small peak counts, a large normalization factor, high background and the worst of all, peaks were not distinguishable from the flutuations in the background. In these cases, cross sections for weak states at those angles were not obtained. To minimize these possible errors, data were treated as follows: For a given state, the preliminary angular distributions at all four energies were displayed on one 4-cycle semi-log graph. The shapes of distribution were carefully examined and compared. If some data points appeared to be off course, they were rechecked for accuracy. Very frequently every datum point in a spectrum was checked for its credit of confidence, i.e., taking all sources of error into account to determine the permissible range of correction. Only points with poor confidence levels were corrected if indications showed this to be desirable and the corrections 42 were required to lie within the limit of total possible error. Usually smooth distributions were normally obtained. But one must not push too far to make the final distribution satisfy his own taste. For in the case of weakly-excited states whose data points were associated with large error bars, any altera- tion of the shape of distributions would be possible. An example is the angular distributions of the first excited state. The distribution of 25 MeV looks different from other three at 30, 35 and 40 MeV. These cross section points were then reanalyzed for many cycles and the distinction between the result of 25 MeV from others was confirmed. Extreme care was taken in the analysis of small angle data because they play an important role in the determination of the spin transfer. Effort was also made to obtain the distributions for composite peaks as accurately as possible so that meaningful decomposition of these multiplets could be carried out (see Section 3.8 and Fig. 3.2). 3.8 The Decompgsition of Multiplets From the knowledge of the exact position of excited states which we have on the basis of Grace and Poletti's spectrum (Gr 66), we know that several pairs of doublets with about 20 Rev separation were seen as single peaks in our spectra. Individual distributions could not be extracted directly from spectra for these states. It was decided that the angular distribution for the composition peak be analyzed first. Then, decomposition was done whenever it was possible. 43 Fig. 3.2 illustrates the decomposition of the doublet at 8.558 MeV. The spins of the component states were tentatively determined by examining the overall shape of the combined dis- tribution and by intelligent guessing. In this case they are 5- and 2+. The experimental angular distributions of 4.48 (5-) and 3.90 (2+) states were used for mixing, with a proportional ratio. The resultant distribution was com- pared with that of the experimental doublet. The best ratio could be obtained by finding the best fit to all distribu- tions at four energies. As shown in Fig. 3.2, theSe fits were very good except at Ep=25 MeV. Aside from the criteron of being a good fit for all four beam energies, the difference in differential cross section at various angles must also be in consistent with the change of peak shape and centroid from one spectrum to another. It was found that, by careful inspection, the change of peak shape for this multiplet agreed with the above analysis. This also provided a way to determine the association of the spin and the excitation energy of the component states. The differential cross sections so obtained were estimated to be accurate to 30%. Similar analyses applied to the doublets at 7.539 and 8.097 MeV. The results were shown in Fig. 3.3 and 3.4. For the composite peak at 7.539 MeV, a fit was obtained by using the distributions of the 3‘ (3.73 MeV) and 4* (6.50 MeV). One may argue that the differentiation in angular distri- butions between £=3 and i=4 states is not significant enough 44 DECOMPOSITION OF I DATUM. POINT MULTIPLET AT — COMBINED DIST. 8.558 MeV --— 8.535 (5") ----< 8.578 (2’) L05- LOE' ‘3 i 5 ED: 35 MeV 2 I- r- - I. b d 2 OJ 5— 0.I .— —.‘. m '- b q B t - . s g : . : V '\.\\ d __ __ \.\:‘\ _ 3: x: 1300' '- 0.0| 1" ‘5 I t = L . - OJE- OJ;- '1 0.0| l l J ‘ ‘ l ‘ ‘ ‘ l J l l l i L 1 1 1 1 1 20 4O 60 80 IOO 20 4O 60 80 I00 9mm) 45.. CY. Figure 3.2.--The decomposition of doublet at Ex=8.558 MeV. da-ldn (mb/sr) 45 DECOMPOSITION OF 6 DATUM POINT MULTIPLET AT -— COMBINED DIST. 7.539 MeV 7.558 (4) 7.53I (3) |.O _ _ _ E = a : E =40 MeV E E =35 MeV : p _ p ‘ I * l I. - -I 0.I ___ .L 1 : E I I ; . I- I- '\.\\ \.\\\ 0.0I . _ j? I ‘ 3 P ‘P . I I 1 L - i . . I 0.0| l 1 1 l L L 1 l l 1 1 L__L L L l l L L J 1 LL 0 20 40 60 80 IOO 0 20 40 60 80 I00 60M (deg) MSU CYC Figure 3.3.--The decomposition of doublet at Ex=7'539 MeV. (kflUn hub/91 46 DECOMPOSITION OF . 0mm pom-r MULTIPLET AT - COMBINED DIST e 09-, MeV 8.090 (2) . "" 80"0 (3’ LC L" LO .- 1 E *\ Ep=40 MeV E 3 L I. 1 0.I :- 0.| r 1 : F : C I I F \ \ i ; ‘? ~~~.§\ J7 . I.0 _ \ I.0 : 1 E E : I c 3 h f 0.I __ __ _ E E : : 1 I. L '1 I- L . 0.0IL.1..L..LJ. W 20 4O 60 80 I00 20 40 60 80 I00 ecu (deg) Figure 3.4.--The decomposition of doublet at EX=8.097 MeV. 47 to allow a definite conclusion to be drawn. It was true that the uncertainties in the differential cross sections of the component states were quite high. However, judging from the smallness of the relative errors in cross sections, the angular position of the maximum, the lack of structure of the distribution, as well as the consistency between the proposed decomposition and peak features, it was thought that the result of this analysis would not be far from the truth. The components of 8.097 MeV were assigned 2+ and 3‘. It should be noted that the experimental distribution of 6.28 MeV state, instead of that of 3.73 MeV state, was used for 3- to obtain uhe best overall fit. 3.9 The Analysis of 6.905 and 6.944 States Grace and Poletti observed a triplet with excitation energies at 6.909, 6.930 and 6.948 MeV. The 6.930 level was seen to be the strongest among this triplet in their spectrum taken at 87.50 at Ep=l3.065 MeV. In the present experiment the level energies were assigned (see Section 3.10) 6.905, 6.926 and 6.944 MeV. As shown in Fig. 3.5, the first and third of this triplet were quite well resolved at smaller angles while the middle one was not seen. At larger forward angles, they were partially resolved and the 6.926 level could be recognized. It can be inferred from this and Grace and Poletti's observations that the differential cross section of the 6.926 state is probably small and its spin may be at least higher than 2. 48 A program was written to analyze this multiplet. Only the first and third levels were analyzed. The program was to find the best fit to this part of the spectrum by superimposing two standard peaks 40 KeV apart. Various standard peak shapes observed in this experiment were stored in the program as options to be selected. The input includes the Spectrum deck and a control card which indicates the approxi- mate centroids of the component peaks, background levels and an option number. The program will search for the heights of the individual ideal peaks, add total counts per channel, compare with the experimental spectrum and calculate a x2. It will also move the ideal peaks one-fifth channel per step on both sides across the pre-set channel for their centroids, to search for the minimum x2. The output con- sists of the area and the centroid of each peak, comparison of total net area and most important of all, a printed graph of fitting. Searches can be repeated by putting in more control cards. The result of this analysis is illustrated in Fig. 3.5. It was found that the fitting was very sensitive to the resolution of the standard peak used. In this result, best fits were obtained by choosing a standard peak with resolution of 33 Kev (FWHM). At laboratory angles equal to 12° and 27°, the quality of fit and the cleanness in the valley suggested that the 49 a,- 40 MeV . _ E,- 40 Mev . 400 ems n2.o‘ .. 300 _ am: 267' g i .v I . l 3 . w . I - am- - (n l— P ‘ z . 8 P o t ‘ r ° “ IOO L r p A b q 0 A l A l 1 I IND ma) IR» 1- '1 b 1 . 5:40 MeV § . . e,=4o MeV . m 200 - - 200 l- - ‘ Z :3 r- 4 O 0 F d T 4 IOO — u IOO - 4 r- J l w 0 I090 noo mo 0 CHN‘NEL Figure 3.5.--Spectrum fits using two superimposed standard peaks for the analysis of 6.905, 6.926, 6.944 triplet. 50 differential cross section of the middle level at 40 MeV beam energy is less than 0.02 mb/sr in this angular range. Hence the differential cross sections for the 6.905 and 6.944 states are believed to be fairly accurate, and the spin assignments for these two states can be made more or less unambiguiously. At larger angles good fits were still achieved, although the middle level started to show up. The angular distributions and the spin assignments of the 6.905 and 6.944 states are discussed in Chapters IV and V. 3.10 Excitation Energies The excitation energies of the observed levels of Ca40 have been measured in previous works (see Section 5.1). Below 9 MeV, every state seen in this experiment was also reported by Grace and Poletti. However, it was decided to carry¢mn;the energy calibration to check the linearity of the data taking system used in this work and to determine the excitation energies for those states lie above 9 MeV. Program FOILTARCAL written by R. Paddock was employed. The input to the program consisted of beam energy, target thick- ness and orientation, detector angle, type of reaction, cen- troids of peaks in channel number and the calibration energies of 51 reference peaks. The program calculated the laboratory energies for the reference peaks using relativistic kine- matics and then made corrections for the energy loss due to straggling through the target. These calculations so far were independent of any knowledge of centroids fed into the computer. Now using the calculated energies and the experimental centroids of the reference peaks as two independent variables, points of reference peaks were located. A least-squaresfit of linear or quadratic order could be drawn through these points. Fixing the theoretical absolute energy, a calculated centroid corresponding to the calibra- tion energy for a given reference peak was obtained. The experimental centroid of the same peak is converted to the observed energy after the calibration. The determination of energies for non-reference peaks is then straight- forward. 4 The calibration energies for reference peaks were 3.731 MeV (3-), 4.482 MeV (5-) and 6.285 MeV (3-) taken from ref. (Gr 66). The results of the calculation are listed in Table III-l. The energy shown for a given peak was obtained by averaging over the results from all but few spectra of each beam energy and again over all four energies. As can be seen in the table, the consistency of the experimentally determined Q-value for every state was within :1 KeV. A comparison with the Aldermaston measurement showed that agreement at both ends of the spectrum (3.732 vs 3.731 and 8.847 vs 52 8.848) is very good indeed. Comparisons with other experi- ments are shown in Fig. 5.1 and discussed in Section 5.1. No attempt was made to calibrate the energies for closely spaced multiplets. Absolute energies were assigned in consistant with all other levels and the separations were taken from the results given by Grace gE_al. It was found that the calibrated energy for a given state was independent of beam energy, i.e., independent of the absolute energies of the inelastic scattered protons. This fact reflected that both Ge(Li) detectors used possessed good relative charge collection characteristics. It is concluded that the linearity of the electronic setup in this experiment was within 0.1% over about 9 MeV differ- ence in proton energies. “munsoc mumoflucfl . 513 msoon.n.om:onm.od m:mno.n.+m:ckn.pfi Gm ommno.nsom:m:n.nfl mmmnn.n.+n¢msn.rs mm onmoo.n.onmmmm.m owmnn.n.+ommmx.n em nm:no.n.ofifinmm.m nmsno.n.+fiflkwe.c «mm nmuoo.n..moomm.m nmmno.n.+cooow.n mm Hafifio.on;aonfi:.o defiao.nu+fiemw:.r *Hm «mmno.n.+mmkmm.m Hmmnn.n..wmkmn.r om mnoon.n.+m:nmm.o muonn.nu+makpn.r mm mscoo.ncommflafi.m msano.n..mmssr.n «em mxsoo.n.+mmmmn.o mu:no.n-.awmnr.n Gm oxsno.nu+mmskm.m mxsnn.n.+am:sr.x mm mmmoo.nuomoo:y.m o::no.nu+mnm:«.x somnn.nn+nmmau.a ukmnn.nu+finman.x finmuc.o.enmuai.u em ommOOQnu+mnm:k.x “nono.nu+mmu:~.a mmsnn.n.+an::~.u ma:no.nu+flnmsn.u cumV(.n.+mm::k.m mm mo:oo.nn+nmamm.m msaor.nu+nmamm.a noann.n.+nknmm.u 3-:nn.nu+«maux.a mmm(n.o.+nkamm.m 4mm mo:no.ns4numfls.a snono.n.+mfinfl:.m :m:nn.n.+ommfis.m mumno.nu.namss.a an(n.o..mHmH¢.a Hm mnmnn.n.4fimfiom.m mimnn.n.+::nmm.m m::nn.n.+wmmmm.a nkmno.nu»os:«w.h semen.o.+«wwom.m om mmmno.nu+:finmn.m mamnn.ncomnmmn.m ummnn.nu+mmcmn.a mwmnn.nu+wo~ru.. :mM(r.o.+anc>.r *ma mnmnn.n..mosmm.k :nkno.n.+emnmm.k ma:nn.n-.smnnm.k wkkno.n..¢mmnc.h mqmvc.n-+amnnm.k we dkmnc.nu.mdmmm.s nnmna.nu+mfla¢m.n :Hmnn.nu+ummmm.n mmfinn.n.+«:c¢p.m 4wmo >mz mmumm >w2 omnmm >mz mmumm >m2 oeumm xmmm .A>mz aflvovmo How :oflumcHEHmumo pom coflumunflamo xmumCMII.HIHHH mqmée CHAPTER IV COLLECTIVE MODEL ANALYSIS 4.1 DWBA Theory The distorted waves theory of direct nuclear reactions and the treatment of the inelastic scattering have been summarized by Satchler (Sa 64, Sa 67). The formulation is based on the transition amplitude T ")llei+)> (4-1) _ ( Dw‘'s are the "distorted" wave functions of the interacting system. This matrix element can be obtained from the formal scattering T-matrix theory using a pertur- bation method (Ma 64). Lectures on deriving the above equation have been presented in this laboratory by F. Petrovich and B. Preedom who also gave the detailed pre- scriptions for the calculation of this matrix element in terms of various types of reactions and specific nuclear models. The transition amplitude for the reaction A(a,b)B can be written as (-)* '+ 9' (+) + + + + - Ifxmémf(kb,rb) xmimi(ka,ra)dradrb (4 2) 54 55 where fa and Eb are the coordinates of the projectile relative to the target in the initial and final state, and J is the Jacobian of the transformation to these relative coordinates. The function x(k,r) is the spatial part of the distorted wavefunction of the projectile. The matrix element is referred to as a nuclear form factor and contains all the information on nuclear structure, spin and isospin selection rules, the type of reaction involved and so on. It should be noted that the operator V and state vector IaA> are written in an abstract basis. Their expansions over the space of a chosen representation are implied. The distorted waves Xéf;(i,§) are the elastic scatter- ing wavefunctions which describe the relative motion of the pair. They are generated from a Schrodinger equation which contains the one-body optical potential. The subscript m'm denotes the spin projection m' of the distorted wave due to the action of the spin-orbit component of the optical potential on the original impinging wave with spin projec- tion m. If an unpolarized beam and target are used, the differential cross section is obtained by introducing kine- matical factors and appropriately summing and averaging over the spin projections of projectile and target nuclei. “a“b 2 kb 1 = (5;;7’ E2'1zsa+I)IZJA+17 MiMBITDW mamb :48 I2 (4-3) 56 where u is the reduced mass of the projectile. The matrix element is rewritten in angular momentum representations + where 5a and 5b are spins of projectiles, and JA and JB are those of the initial and final states of the nucleus. The rest of the development of the DWBA theory for direct inelastic scattering consists of arriving at analytical expressions for the transition amplitude. This involves two stages of multipole expansions, namely 1) The multipole expansion for the above matrix element into the transferred angular momenta (£,s,j) representations. In analogy to Wigner- Eckart theorem, the transition amplitude is expanded in terms of "reduced" amplitude B 23amb . 2) The partial wave expansion for the distorted waves x to obtain explicit expressions for the reduced amplitude. These expansion treatments put the DW theory on a formal and elegant mathematical foundation. Detailed discussions have been given in previous references (Sa 64). 57 4.1.1 Ca40(p,p') in Collective Model For the Ca40(p,p') reaction, simplified expressions for the form factor can be obtained via several approxima- tions. The interaction considered here is assumed to be 1) local, therefore the "zero-range" condition is satisfied automatically. 2) static (no time dependence) and central, each term of the multipole expansion of V(;,aA)= 2 (-1)3‘“v £53,“ (A’r)Tlsj,-u(fl’a) (4-4) 2'5er being a scalar product, where A, a denote the internal coordinates of target nucleus A and projectile a respectively. The spin of the ground state of Ca40, J , is zero, so j=JB. The spin 1/2 of the proton allows the transfer spin 3 to be 0 or 1, thus + -+ One also finds that in a given transition, possible values of l, s, j are limited. To take j"=3-, for example, there are only two multipole components, (303) and (313). For the special case s=0, the form factor Glsj becomes G£(r)=/2§;:T (4-5) which is used in the following collective model studies. The microscopic model descriptions for the scattering from the odd parity states follow different approaches as pre- sented in Chapter VI. 58 4.1.2 Vibrational Collective Model It is well known that the nuclear collective model has been very successful in explaining the strong transi- tion observed in inelastic scattering. This model assumes a non-spherical potential well V which induces inelastic scattering to low-lying collective vibrational or rotational states. The nuclear deformations modify the average field on a macroscopic scale as felt by the projectile due to the short range nature of the nuclear force. The devia- tions of the average nuclear field from spherical symmetry are described by the theory of Bohr and Mottelson. A treat- ment of this potential in the framework of DW theory has been formulated by Bassel et al. (Ba 62). The spherical potential is just the optical potenial, hence the deviations from spherical symmetry can be obtained by expanding the potential in a Taylor series abOut R=RO - _ _§_ _ U-U(rfl%9 6Rdr U(r R0) + . . . Retaining terms to lst order in GR, one finds that V: -6R d U(r-R ) d? o ' In the vibrational model, the nuclear surface deformations are defined by * M 59 The distortion parameters a are assumed dynamical and LM capable of creating or annihilating phonons of angular momentum L with z-component M. The nuclear potential V is now V=R [—90 -R0 )1 Y“ (6 ¢) 0 dr L, Ma LM L The multipole component VLM is then \7 =iR'R [-3 U(r-R )]a* LM 0 dr 0 LM' * LM in terms of usual boson creation and annihilation operators The dynamical deformation parameters a can be expressed * L . . bLM and bLM for 2 -pole osc111ation 41w fi=(§——>1/2[bgm+<-1)M bLM] where fiwL is the energy of each phonon and CL force parameters. For an even-even target, JA=0 and no initial is the restoring- phonon exists, then =(§——)1/2. If no spin-orbit potential is included in the optical potential U(r-RO), then =-iLRo[dd U(r- R0 )JBVib (4-6) LMIJA where afiw BVib-[(2L+l)7——]l/2 60 It can be seen that the form factor has the game radial shape as %; U(r-RO). This means that in this simplified model, the detailed nature of the nuclear structure is ignored and the total effective interactions are limited into a few standard types of form factors with the interaction strength to be‘extracted by comparison with the observed inelastic cross section. 4.1.3 Vibrational Model Parameters and Reduced Transition Probability The Hamiltonian of a vibrator having dynamical arameter a is P LM _ _ M HL §( 1) (BLGLMGL,-M+CLaLMaL,-M)' where BL is the "mass transport parameters" and CL pre- viously defined as the "restoring force parameter". In terms of the "observables" excitation energy E and the L "model dependent" deformation 6L=BLRO,BL and CL can be found by 2 _ 2 2 (BL/n )—1/2(2L+1)(RO /6L )(l/EL). (4-7) _ 2 2 CL-l/2(2L+l)(Ro /6L )EL. The reduced transition probability for electric excitation L O I O O O of a 2 -pole Vibration in an even—even nucleus is given (Ow 64) by 61 6 2 B(PP';0+L)= {291% }2 i=5. (4—8) 4nRO R0 The results of calculated B(pp') are often compared with the single-particle estimate in Weisskopf unit, i.e., = 'o ...; Gsp B(pp ,0+L)/BSP(EL,O L) where Bsp(pp';O+L)=[(2L+l)/4w]e22, and 2 is calculated using a uniform change distribution. The value of GSp measures in some sense the "collective strength" of the state. It is also of general interest to compare the B(pp')'s with two sum rules. The first is the non-energy-weighted sum rule (La 60). NEWSReg Bn(pp';0+L)=(eZZ/4n). where the sum is over all states with same spin L. The second sum rule is the energy-weighted sum rule (Na 65). = - " EWSR §1(En EO)B(pp ,L+0) =Zze2Lfi2 W(2L+l) 2 where AM is the mass of the nucleus. The results of calculations for these vibrational parameters, B(pp')'s and quantities of comparisons are presented in Section 4.5.2. 62 4.2 Optical Model Analysis In order to obtain parameters for the calculation for the distorted wave x, the angular distributions of elastic scattering were analyzed. The optical potential used in this work was as follows: U(r)=UC(r)-V0f(x)+(fir-T-:-C—:) ZVSOG-mil; gfflxso) . d , -i(WO-4WD d§’)f(x ) where Uc(r) is the Coulomb potential due to a uniformly charged Sphere of radius Rc=l'25 Al/3 and 2 U 2.2.—e... I r>R c r — c Ze2 r2 =fi—(3'7‘) ' riRc' c R C The factor f(x) is of the usual Wood—Saxon shape _ l/3 _ x —1 _r ROA f(x)—(l+e ) where xe a . The parameters which enter the DWBA calculations were determined by fitting the calculated cross sections from this potential to the observed elastic data. The search code GIBELUMP* was used to vary the parameters. The criterion for a fit was to minimize the quantity: * Unpublished FORTRAN-IV computer code written by F. G. Perey and modified by R. M. Haybron at Oak Ridge National Laboratory. 63 w° I I II I II I II I II W0 I II I II I II I II I “Ca(pm) 4OMeV ‘°Cn(p,p) 35MeV V°=44.5| MeV v,=46.42 MeV W,= |.7l MeV W.= 2.37 MeV IOL- Wo= 4.42 MeV -‘ Io~ W,= 4.l7 MeV " )6: 4.28 )6: 6.87 2 OR . OIIJJllllllLlllg o_.lllllllllJJll O 40 00 I20 0 40 BO 120 9mm”) 9cm.(d99) W0 I I IIII In I‘II I II w° I IIII II TIVfi I II I ‘°Ca(p.p) 30Mev ‘°Co(p.p) 25Mev V.=47.86 MeV V.= 48.92 MeV W.= 2.40MeV W.= 2.IOMeV Wo= 4.l8 MeV w”: 4.07 MeV IDI- j IOI- 8 >6: |.94 x: 3.60 _O'_ . Oh . | r— —4 I I— —1 0., lLlllLLlllllJ 0., llLlLlLllllL O 40 00 IZO O 40 BO IZO ec.m.(deg) euanBQ) Figure 4.l.--Optical model fits to the experimental elastic scattering results at Ep=25 to 40 MeV. 64 [Oex(ei)_cth(ei) 2 1 A0 (07) ex 1 where N is the number of data points, Uex(0i) is the observed differential cross section at the center-of~mass angle 6i and oth(ei) is the theoretical value at Bi. The relative uncertainty Aoex(0i) was taken to be 3% of oex(6i) for all data points. The geometrical parameters (r0 and a) for various components of the optical potential and the average spin- orbit strength (V80) were taken from the analysis of elastic scattering and polarization measurements for 40 MeV protons on eleven nuclei from 12C to 208Pb. (Fr 67). The remaining parameters searched were V0, W0 and WD. The results are listed in Table IV-l. These parameters were used for the DW calculations presented in this study. The elastic data, in ratio to Rutherford scattering, and the final optical model calculation are shown in Fig. 4-1. 4.3 DWBA Calculations The DW calculations were made using a FORTRAN-IV version of the Oak Ridge computer code JULIE (OR 62, 67). * The program has been adapted onto the XDS, 2-7 computer 'k Unpublished Sigma-7 program description on JULIE, Cyclotron Laboratory, Michigan State University. 65 TABLE IV-l.--Optical Parameters. FR = 1.16 F, aR = 0.75 F PI = 1.37 F, aI = 0.63 F FSO = 1.064 F aSO = 0.738 F VSO = 6.04 MeV 2 Ep(MeV) V0(MeV) WO(MeV) WD(MeV) x 25 48.92 2.10 4.07 3.60 30 47.86 2.40 4.18 1.90 35 46.42 2.37 4.17 6.87 40 44.51 1.71 4.42 4.28 and stored in the computer's file under the timesharing Janus system. Typical running time was about 1 to 2 minutes per case depending on the scope of calculation involved. The input consists of three major parts corresponding to the elements in the integral of the transition amplitude (Eq. 4.2) namely, the form factor, the entrance channel (incoming DW) and the exit channel (outgoing DW). The form factors used for collective model were complex. The real part was calculated by JULIE whereas the imaginary part was external numerical input. Options 2 and 3 as in SALLY (pp. 64, OR 62) were used for i=2 and £=3 respectively. For 2 larger than 3, the value of hi (pp. 42, OR 62) was set to zero, i.e., no Coulomb potential was included. Since 66 the spin-orbit potential for form factor calculations was not provided by JULIE, only G was computed. In other 202 words, spin flip was not taken into account. The imaginary potential was just the first derivative of the imaginary part of the optical potential with respect to r. The input deck for JULIE was provided by the program DEFABSORB written by B. Preedom and K. Thompson. The input for the entrance channel was essentially the set of optical model parameters listed in Table IV-l plus controls over the option of potential used and the maximum angular momentum of the partial waves included. The optical parameters for the exit channel used depend on whether the Q-value effect was considered or not. Fig. 4.2 is shown to summarize the general results of the calculations for 2:2 to i=8 and for energy dependence as well as the Q-value effect. For i=8, spin-orbit term in optical potential can not be included unless j=£ (Table 1, OR 66). In order to see the effect of the spin-orbit potential on the distribution, calculations were made with and without this term in both entrance and exit channels for the case of £=6. It was found that the effect is small except for 25 MeV as illustrated. The normalization between the output of JULIE and the experimental distribution, for (p,p') and complex collective model using options F6=2 or 3, is (3'7 m .>w2 ov 0» mm" m pm mfla 0» NNa HON .m.m HOUOE w>HuomHHoo moans mcowumasoamo daze on» no mumafismuu.~.v gunman ......Ju 88880.8 8... ecu 83. 92:30 :3 9.4 E 58 2. 89d... 82 1:228 :98..sz . 0' . On . s . some) 00 . 9 0n . On 440 g 5... Egg >938 um: ..... hzgkwaamq 02 III wmwhgfia 4407—00 42230 :xw 2. ommmeZOu wouudw uZZ> O .838. O” 8 8 8 9 ON b. V A j A H 3.. on we "I \\\L to— \ r \\ . v )5 8 \\ A 1 I \ A. u ,7 \ v w /, .\ . wl l.“ \\ o. i 8 \ \ v 4 \\ . \\ L / \\\ k v V r w ,, .. n x . I I] / \ / I/l!\\\ \\\\ ”— I v I, \ A f I . I . i 0' x, x v / \\ A . xx \\ u r // \x L :0. v r m . 4 . H d8: 38 3o . v on H ah v 83 u Iwr‘rrv V V TYVVYV w vav v ......J. gasses IIIIIIIIIIIIIIJ I // ”v v . lu\>/ v I . I: on / \XL . is on / H n / \. . n x \ . v \I / \\ u / \\ mu I/ z/ \\ l b. V I, \\ IllI“ L (II \\x is / .V . acIrIIII /I\\ I v In \\\ v 3 9 II/ A w I . I L / \ . , \\ . ./ \\ n I, \x g .I z \\ I b. mu / \\ I //'|\ 1’ \\\ v v /|\ v s . a . w ... . 4R5} JJOU 30 v .683 4.50 30 V -O.B~ 800 A w ah.3~ 800 . P r . . p ’ F b, hr ’ b p ’ D D ’ i? 08888994. m u I .600! joy :5 .... 3 88 thth P .P P } ALA“ ‘ V O. (ammmm) 0W" (3110f)(fl/¢M) lip/OP 68 l 2JB+1 2 “ex!” = 315i? 217:: 23?: BL 0L (JULIE) where L is the tranferred orbital angular momentum. For an even-even target JB=L and JA=0’ the above equation becomes 62 G(exp) = 50i4 0L (JULIE) The differential cross section scales in Fig. 4.2 were taken directly from the JULIE calculation so that consistency was maintained throughout. To extract the deformation parameters BL, program SIGTOTE (Th 69) was used. This program compares the total cross sections G(exp) and 0L(JULIE) within the angular range of this experiment according to the equation ' 6f _ 2 6f do . 0(exp)le' - fifei 35 (exp) Sine d8 1 and then calculates the B The code also commands a L' computer routine to plot the collective model fit on 4-cycle semi-log graph. The deformation 6L is defined as BLRO where RO is the real radius of the target nucleus rRAl/3. For 4OCa, R0 = 3.96 fm.was used throughout. 4L4 Representative Results of Collective Model Fit Fig. 4.3 shows the results of the representative 2+(3.9oo MeV), 3‘(3.732 and 6.281 MeV), 4+(6.502 MeV) and .mmumcm Econ 09 pommmwu zufi3 cofluonfinumwp Hmucoaflwmmxm on» no manna mo GOAuMflHm> map 3Com muumm Home: was .Amm>uoo pHHOmV muwm ampofi m>wuooaaoo pom mcowusnauumflo Hmucwaaummxm mo mpaomou HMUfim>BII.m.v muomflh 3.330 3.3-w zinc 3.388 isle 08880.98 DINIRWIm'sovON 0888898, 888888 88880.8 H >§.m~.o..u..n # igmfim U h had. our: . C. a 300' .8389 70 5-(4.487 MeV) states. Except for the 6.502 level all other four are strongly excited. At the top of each column of Fig. 4.3, the energy dependence of the shape of the angular distributions are illustrated. It is seen that the structure of the distribution becomes more pronounced as the beam energy increases. At Ep=30 to 40 MeV the shapes of angular distributions with different L-transfer are quite distinc- tive one from the other. The lower part of the column indicates the comparisons between data and collective model calculations. The results of individual states are dis- cussed in the following paragraphs. 2+(3.900 MeV): The angular distributions of this state have a unique shape. The differential cross sections peak at small angle and dropping off somewhat slower up to about 30° than they do past 30°. A flat region occurs at around 50 degree at Ep=40 MeV and moving out steadily to around 70 degree at 25 MeV. Following that are a fast descent and another flat region again. This feature distinguishes the 2+ from 3- and l- and provides a positive method for identifying the spin of this state. The collective model predictions are very good specially at 25 and 30 MeV. At 35 and 40 MeV good agreements are still achieved except at large angles. The success of the model in this case is that not only the shape of the empirical distributions at four energies are reproduced, but also the relative magnitude as well, as revealed by the constant of 62's. 71 3-(3.732 MeV): Similar to the case of 2+, the shape of the angular distribution changes smoothly as the beam energy varies. A maximum occurs at about 25 degree, but its magnitude decreases almost 40% as Ep drops from 40 to 25 MeV. The quality of the collective model fit for this state is comparable to that for 2+ state. The deformation 53 (1.35 fm) varies only about 5% among the four proton energies. Again the energy dependence patterns of the calculation coincide with those of the experimental observations. 3-(6.281 MeV): As can be seen from Fig. 4.3, the shapes of the angular distributions of this state appear somewhat different from those of the 3.732 MeV state. The maximum is also at about 25 degree but here the magnitudes are approximately constant. At Ep=40 MeV there is a second maximum located at about 62 degree which is washed out as the beam energy drops to 25 MeV. Consequently the energy dependence looks dissimilar to the 3.732 MeV state. Sizable discrepancies between the calculations and the data at large forward angles can be seen. However, the relative ratios of the total cross sections under the angular range of the experiment do not differ appreciably between the results of theory and experiment as indicated by the small variations of 6 (only 13%). 72 4+(6.502 MeV): The angular distributions are similar to those of 3.732 MeV state except that the maximum is shifted to about 35 degree. At 25 MeV, the distribution of this state is not clearly distinguishable from that of either 3- or 5-. The spin identification has to be made by using every piece of evidence available, namely the consistency in the angular positions of maxima, the collective model fits and the comparisons with the results of 3- and 5- at every beam energy. Again the deformations obtained flutuate only a few percent. 5-(4.487 MeV): The dominant characteristic of the experimental distributions of this state is the lack of structure as a function of angle. The collective model cal- culations underestimate the cross sections at both small and some large angles but overshoot around the maximum at about 45°. The predicted increment of the magnitude of the maximum is more than 70% from Ep=25 MeV to 40 MeV, whereas the data show less than 10%. On the other hand, the deformation decreases only 10% as for the case of second 3-(6.281 MeV). 4.5 General Results of Collective Model Analysis In this section the_general results of the experi- mental and the collective model analysis are summarized in terms of the L transfer assignment, nuclear deformation and reduced transition probabilities. Comparisons with other experiments will be presented in Chapter V. 73 4.5.1‘ L Transfer AssignmentS'and Nuclear Deformations The representative angular distributions discussed in the last section were used heavily as standards to assist in the determinations of the L-transfers to other states. It was found that most of the angular distributions with the same L at the same energy resemble each other in shape. Distributions revealing possible differences in micro- scopic structure and reaction mechanism were also noted. Since there are four distributions for each state to be compared with the standards, the ambiguities in determining the L-transfer for a given state were minimized. The L assignments to the components of a doublet were obtained from the decomposition method (see Section 3.8). The high spin states having L=6 or L=7 were identified by finding the best fit to the experimental distributions with those from calculation using L=5, 6, 7 and 8. Distorted wave collective model calculations were done for every state with apprOpriate adjustments for the Q-value effects in exit channels. Nuclear deformations were then extracted using program SIGTOTE (Section 4.3). The L-value assigened and the deformation parameters along with other physical quantities are listed in Table IV-2 to IV-5. The experimental data, the collective model fits, and the standard distributions are shown in Fig. 4.4 to Fig. 4.8, where the solid curves are collective model calculations and the dashed curves show the shapes of the standard distributions. dc/dn (Mb/8f) dc/dn (Mb/t!) O I '°r‘\«\ . ? fi' ‘1 f ‘V T 1 V ”Co (p.9’) E. - 3.900 HOV A ALAAL . Anni .4 4o qu aeoazer 5 q 4 35M‘ /"\ 8:042“ \ _‘ 1 d 4 °’ - 1 4 4 ‘ 30”“! 8‘04 32F 4 > \\ 25 W 4 i Ky 8=0420F 4 A A A zo‘c‘o {o‘oofio‘oWzoA > I V Y r V W I f 1 = A wam0 5 En 7.290 HIV 1 ‘4' 1 [N 25m: 820203 F 720 Figure 4.4.--Experimental distributions of i=2 states and collective model fits (solid curves). 74 “Co (p,p') 5.9 $240 Mov A A‘ALAA F t \ r I A LALMJL non-v ‘ a-ozzsr 1 l 1‘. 35M .. $02236?fl F '5 ; : b 1 b 1 > 1 r 1 I L > 4 > -+ ”to T V Y 1 —fi T V ‘V Y Y ‘0 : a ('09.) ‘ I 5.4.905 mv . LO :- 1 .f, f r 1 » 40 NOV * 3:0447F |.0 E- ‘1 f I F I - as «w < 0'0424F W F" ‘: J? I > w II 0432F l0 5- 1 ' I I 1 , 25W . 8.0440F OJ ‘ 4 A A A A -A—A-—I—l—J 20 40 60 N 00 I20 ‘9' r d 1 houses“; 4 35“ ... a-ounr 3 30““ °" :— moms-i L' b 1 " 5M * B'OtTO QOIL A A A . A A A A A A 20 40 60 00 I00 I20 The dash curves are those of 3.900 MeV state. 75 .mumum >mz mmh.m mo mmonu mum mm>uso swap one .Amo>nso oHHOmV muflm HmUOE m>wuomaaoo can mmumum mna mo coausnauumwo HmucmEHnmmxmll.m.v musmflm 8.... 11.... .83.... 1.3.... 9.88388 8.88898 88880.8 V “.329. 2......“ u :8 “I Lsfloi vizvob A. . ion v T35.» 33%» . >18 v 138.» , 3:29.... W >38 " V v \ v v ‘ . . igvoou . 33513.6 . «ha. 80' v a}. 80' W tn..- 800 >>>>>>>>>>>> } r f b P ?(P h (F L! h r h S h > L > p h h P p r b F 3.8a +8-8 mac‘s m-a. 2.8 s s s a 18»... A .inu . W 0.. "Lagos ‘ a "333/ .. u 2.38,... a r38 >03 Nnhn ..u fi a... 8.. i b b p h b i n . . >¥PP}}(P>>>>>5 0. (MM moo do!“ (mu/u) GORMJMMhfl L8 L0 E “Co (pm’) I E.- urouuv A AAAA 40M beI27F 30W \ 8-0147F 1 ‘ ‘ ’ 4 25'”! a-onez F 1 A A A A A A A A A U I r V I V T V V ' fir WauJ) E‘I‘LQZI mv A 44;. '3 40M ‘ asozesm - q ”Md! 1 O‘OZBSF 1 30M : 8-03I3F j 76 A_._ F 0.I v rvvvvvv' O‘Ol LO “mama . E. -6.502 mm 1 j, ‘ I A It‘llll . Q 30 m ”0'7‘F '3 1 . 29w: '. FOIOZF‘ ZOOOQNWIO Y Y 'YT‘Y' I 'V'I", I ' r v v r w .9 . 3 Co (9.9') : EfOJGI W ‘ 0.I I D 0.I Tyyfrvvvrtvi' ”any: E. 0 7.454 HOV A A‘AAAAI A A AAAAA‘I 40 “I WIMP AA l AAA]... 0.: am] 0.I ’ 0.0! Figure 4.6.--Experimental distributions of L=4 states. curves are collective model fits. those of 6.502 state. Dash curves are Solid 77 .muwum >m£ hmv.v mo moon» mum mo>noo nmwo one .Amm>uso odaomv made HoUOE 0>Huomaaoo odd mouuum mua mo soflusnfluumdo HoucmfiwummeIl.h.¢ whomwm u>u 3m! '0 save 8. 8. 8 3 o. 8 w .u .00 1. “3m. 0“» J . 2.5030...” v )0: on // A H xr U m ,J \.\\. m . 109?». 7 g -.o >§w8mnu 7 Y >22 mm 1:7 “v v */+/ n 7 w n 7 )..u / / \\\\ u v emit a +,, frg 1._.o >§nmom~w , w >9: mm j m. . a e t l n H W 1” S “V u 1. _.o l vuamofi v ion l 8.3.)» >22 on ITYYVU ‘ Y unmwonn >039 r 1777177 I I )0! 0.2.. .u .... 3 82 P b b b b r b h > P b P 7711' V V A 0.. 1vvvvvv v ‘T V Ivv'vvy v ff Ivvvvvvv vav ' . e38; / 38.:o 098.0083 ‘ ‘ ON AAAALL L4 1 uQNVOuo >01 n“ . 3. noon 0.» >i on .4, . ‘LL*+ >03 on 2) >3 3.» ..u .uatds > ¥ b P F F . h p b b .40 381$ 'TT 1'— I'V'V Y ITYYV' V unomona inn 1 umnmouw . >i8 L4] >3. 53:26 is: 89 ' > h h » .P F > O. ("N”) '9/‘09 78 .Amo>uoo oflaomv muwm HmoOE m>fluooaaoo can .mmuoum hue com mua mo moowusnauumflo Hmucuawummxmll.m.v onsuflm as a... size 080900898 .0: 3.0....“ in» “Jill A 5.0 1 k 500"» '30? 'T 'V—r 2.2533. . . C. .3 83 r b b \P h b h h b b P p [111111 I ‘_.0 $80?» "Li! nN . ..O 5 (Is/aw) 09/09 ML .83 3a 0809 T new?» .3233...“ n >228 I w... as?» . 2:8..le I IVYT I w... at on» iguanu I r >02 on .... cm. .036 + . chmdu 3’02 OV TWTY V V‘TT Y I TTTFT I fir - A \ a. . Q 880? d d >33» .3 : (”I”) oW‘W 79 4.5.2 Reduced Transition Probabilities Having made the L-assignments to all excited states and extracted the corresponding nuclear deformation parameters 8 one can then make the calculations for the reduced transi- LI tion probabilities B(pp'; 0+L). The values of B(pp') obtained from this method are model-and parameter-dependent, namely on the quantities 2L-2 r < >, B and R . Assuming that for a given excited state L 0 its angular distribution is well fitted by a collective model calculation, then the deformation parameter BL extracted will not be subject to high uncertainty except in its model dependence. The remaining factor which is in question is because of its strong dependence on the transition density p(;) and consequently on the parameters within. Gruhn g£_§l. (Gr 69) have investigated the sensitivity of the B(EL)(p,p') results to the parameters of the tranSition density for 58Ni. They also compared the calculated B(EL)(p’p.) using three different models of the density function. Their finding is that when the non-uniform density distributions determined by electron elastic scatter— ixmgare used, the result of a calculation for (p,p')B(EL)'s are most inconsistent with the (EM)B(EL)'s for the high multipolarity transitions. Gruhn gt_gl. suggested that if a uniform-density distribution having a radius equal to the Fermi "equivalent" uniform—charge-density radius be used, qualitative agreement with the (EM)B(EL)'s was recovered. 80 It is for this reason that this prescription was used for this thesis to calculate the (p,p')B(EL)'s for Ca40. Indeed, very good agreement was found when the results were compared with those of (e,e') and (p,p'y) experiments (see Section 5.3). The quantities B(EL)'s, G(sp)'s, BL/h2 and CL were calculated using the program VIPAR written by C. Gruhn and K. Thompson. A Fermi equivalent uniform- density distribution with r0=l.33 Fm (R0=4.SS fm for Ca4o) was used. The results are listed in Table IV-2 to IV-5. 81 000.0 000.0 m0.uom:.0 mn+moom.0 00.0 00.mmm0.0 0m0.0 mm0.0 m mm0.m 000.0 :00.0 m0.mmnm.0 mo.mmmc.0 m0.0 uc.mmmm.0 300.0 000.0 0 300.0 m00.o 000.0 00+m000.0 30+mm0m.0 mm.0 mo+m00m.0 000.0 0nn.0 0 0:0.0 000.0 000.0 m0.m:mm.0 00+m000.0 mm.0 m0+MOM0.0 000.c 000.0 m 0:0.0 000.0 000.0 m).m.m0.0 m0.mmnm.0 .m.0 m0.mom0.e 050.0 0.0.0 m w~0.0 000.0 000.0 m04wmmm.0 mn+mm0:.0 00.0 00+mmm0.0 mw0.e 030.0 m mmm.w 000.0 000.0 :0.mmmm.0 00.mmM0.0 00.0 00.M0\u.0 m:a.0 mm0.0 m m0:.w 000.0 mm0.0 :0.mnmn.0 m0+mm00.0 0u.m m0+m00m.0 mmm.0 cnc.0 : 0mm.w 000.0 000.0 m0.M00m.0 00+m~mm.0 mm.0 ~0+mm0«.0 0m0.0 umn.0 0 000.0 000.0 000.0 m0.mmm0.0 m0+mcmm.0 mm.0 m0.m000.0 «00.0 nmr.0 m 000.0 0m0.0 000.0 .0.mm0m.0 00.mm.0.0 mm.0 m0.m00«.0 «mm.0 0.0.0 . 0mm.n 000.0 «00.0 ¢0+m¢m~.0 00+mmm0.0 mm.0 00.m~:m.0 «00.0 mma.0 m mom.n 000.0 000.0 mo.mm0m.0 m0.m0mm.0 00.0 m0.M000.0 .00.0 ..c.0 . 000.0 000.0 000.0 mo.mm.m.0 m0.mnm..0 00.0 m0.m00m.0 000.0 $00.0 « 000.0 000.0 .00.0 m0.wmnm.0 00.mmmm.0 04.0 .0.m.mm.0 «00.4 wmc.0 . .m..n 000.0 000.0 mo.mmm..0 00.mnmm.0 00.0 00.mmom.0 000.0 0ar.0 0 000.5 000.0 000.0 m0.mm00.0 m0.m0~m.0 05.0 00.mmm..0 mm... «00.0 m 000.0 m00.0 mo0.0 :0.mmn0.0 m0.munm.0 mm.m m0.mcm0.0 0:..r m00.0 m m00.e 000.0 000.0 mo.um00.0 m0.mmmm.0 00.0 mo.momm.0 .00.” «.0.0 0 0.0.0 000.0 mm0.0 .0.m.n..0 00.m000.0 04.0 .0.M0.0.0 007.0 .nc.w m nam.o 000.0 m00.0 m0.w00m.0 m0.mma..0 wm.0 .0.m0mm.0 000.0 0...0 . 000.0 0.0.0 .00.0 .0.mmnm.0 ~0.mo0~.0 mm.m .0.mmmm.0 40..“ mar.0 m 0xm.o 000.0 m00.0 mo.mmm0.0 m0.mmm..0 04.0 .0.mnmm.0 «00.0 or..0 0 000.0 om0.0 000.0 :0.moom.0 00+mm0n.0 mm.m 00+m:~m.0 num.0 «no.0 m m0n.m m00.0 000.n m3+msnm.0 :0.m000.0 0m.0 .0.mmms.0 0m0.( «00.0 e 00m.m m00.0 .00.c m0.m0~0.0 m0.mmmm.0 «0.0 m0.m000.0 000.0 omc.0 m 0.m.m m00.0 000.0 m0.m«00.0 m0.mcmm.0 «0.00 00.0mmm.0 00«.r n00.0 m 0m... mm0.0 m00.0 .0.m000.0 00.m.«a.0 00.0 00.m.m0.0 .0... .00.0 m 000.0 00m.0 000.0 00.00m0.0 m0.mm00.0 om.mm 00.m0.m.0 m00.0 «00.0 m mmn.m mmzmz 0m :65 0. 3.0.5 000...: E 3sz 0q+o«.dmvm camAAmv go «exam Ammvao 0.dmvm 00 00 a xm .>mz ovnmm.0.m.mvovmo .mnmumfimumm Hmsoflumunfl> w>HuUmHHOUII.NI>H mqmde 82 000.0 000.0 mo.0000.0 00.0.00.0 00.0 .0.000¢.0 nn0.0 n...0 . 000.0 000.0 000.0 m0.m:00.0 00+m0nm.0 30.0 00.m0mm.0 :00.0 mac.0 m m:0.0 000.0 000.0 m0+m000.0 00+m000.0 00.0 00.000m.0 000.0 030.0 m 000.0 $00.0 monon mo;mmm:.n mn+mmm:.0 mm.0 .O+mmHm.n "m0.u cmron m Ohm.m 000.0 000.r m0.ummm.0 00.m¢0:.0 00.0 30.0:0<.0 r00.n xw(.n : 030.0 0mn.0 0m0.0 mnomwn0.0 wnom000.o 00.0 o.m«mc.0 unm.r <¢r.n 0 000.0 000.0 00 .0 m0.mmmm.0 00+mmmm.0 «0.0 00.m00n.0 n00.( om(.0 0 000.0 000.0 000.0 00.000a.0 :0.mnma.0 m0.0 00+m000.0 000.0 :0r.0 0 000.0 000.0 000.0 mn.m0m0.0 00+mma0.0 00.0 00+m00\.0 ’00.: umr.0 0 030.0 000.0 mwo.r mo.m00:.0 mnomnnm.0 m0.0 00.m0m0.0 «m0.: bar.» m mmn.m 000.0 000.0 00.000:.0 00+m00m.0 mn.0 ~0.m000.0 «00.0 rsr.x 0 300.0 mon.0 000.0 00.m000.0 30+mn0m.o 0m.0 00.MM00.0 000.: 00v.” 0 030.0 000.0 000.c :0.m000.0 00+m:00.0 mu.0 mv.mn:0.0 0:0.“ 30:.” 0 030.0 000.0 000.0 m0.m000.0 00.m00m.0 mm.0 m0+mnm0.n 000.. cm(.u m 000.0 .000.0 000.0 mn.m0mm.0 00+momm.0 04.0 mu.m~0c.m 000.( a:r.m m mmm.m w«n.n 00n.n :0+mmmm.n mn+mnr0.o 00.0 ro+mflsa.n 7:0.u omL.n m m0:.x 000.0 000.0 ¢0+mmmk.0 00.m:00.0 0.0 00.mw0n.0 000.. vnc.n a 0mm.x 000.0 won.( moomnan.n mn+mnnn.n 00.0 ~o+mmon.z 4:0.< :m(.u 0 «10.0 000.0 ~00.( m0.mmvm.n room0mm.o 00.0 00+m0mn.0 ”m0.: but.“ 0 000.0 000.0 000.0 mhommm0.0 00.mamn.o 00.0 mu.m010.0 440.- rmr.u m 0r0.« :mr.0 000.0 :0.mmmm.0 ~0+m0:0.u mm.0 00+mvan.0 rxz.- oq(.« : 00¢.0 000.0 «00.0 :0.m000.0 00+mmm0.0 00.0 mn+m~:~.0 «00.: wm«.a « mwx.n 000.0 «00.0 mn+mm00.v mnomahm.n m0.n mn.m:00.L «00.x «4:.» 3 mnw.~ «00.0 000.0 m0+mxm0.n mn0m0:m.o 04.0 mo+mnmw.0 «00.: rr-.n w 0mu.m 000.0 000.0 m0.mx00.0 0.000m.0 04.0 .0.mgmn.0 .00.r ha-.0 3 30:.0 000.0 000.. m0.0mm..0 00.0000.0 00.0 00.000m.p 00(.- 0u‘., 0 000.0 000.0 000.0 :0.m000.0 00.00r0.0 00.0 00.m«y:.0 uxr.. 04(.4 m 000.0 000.0 00c.0 .0.0000.0 00.000..0 00.0 00.0000.0 :0..0 co..: 0 000.0 m00.0 000.v :0.m0mm.0 00+mu0m.0 m0.0 wn.mnmq.m 300.0 amp.” 0 010.0 mm0.n :m0.r ao+mmm:.0 00+mmon.o 00.0 +0.m¢00.r "00.0 unc.. m 000.0 000.0 mnn.( m0+m0m0.0 mn+mhw4.u 00.0 00+M0mr.n 000.: 04..“ 3 m“:.0 0:0.0 :m0.o 30+mmmm.n mn.mmm0.u :¢.m sn+m¢mn.0 x»..« nor.» 0 0rw.w 000.0 000.( 00.0000.0 00.000n.0 04.0 00.000... ca0. n:(. 0 000.0 wmn.o m00.L :00mmnm.n ”nomm00.o am.m mn+mmm\.n mam., .\(. m M0m.m m00.0 000.0 m0+m00m.0 mn+m¢0m.0 mm.0 00+mvma.m xm0.« or:.; 3 00:.0 m00.0 000.0 m00m0m0.0 mn+m000.o 00.0 00.m000.0 0.00... ant.“ m 0:0.m nu0.n 000.0 mh0mm:\.n mn+monm.n 00.00 no+mn04.m amh.y 0(0.o m 001.: mm0.0 mmn.r anomm00.0 mn+mawm.n 00.m m0.mmm0.0 4m:.\ nC¢.m m 000.m 000.0 000.0 00.0040.0 00.0000.0 00.00 00.0m00.0 «30.0 u0o.0 0 000.0 00302 00 :65 0.. $sz 0.005 E EmE 00+00.mmvm 0020000 go 0:\qm 000000 0.0000 00 00 0 xm .>mz mmnmm .0.m.mvovmu .muwwwfimumm Hmcoflumunfl> m>0uomHHOUII.mn>H mqm+m:dfi.0 «ma.r ¢:L.0 : mmm.m 000.0 ~00.0 m0+mmma.0 ~0+mo:m.0 0:.0 mo+mnmm.0 meg.“ 000.0 m ‘Hmm.n 000.0 m00.n m0.mo:m.0 00+mmm:.0 mm.0 30+m03o.0 ana.r v¢0.0 : ¢m¢.u 000.0 000.0 m0+mmmm.0 m0+msmn.0 fifi.0 H0+momm.0 000.0 Mmr.0 m 00m.m mm0.0 mH0.0 30+mmm~.0 mn+mmma.0 mm.m m0+m0m¢.0 “mm.c «no.0 m 00H.n 000.0 mm0.r 30+mfim«.0 m0+mH0:.0 nfi.m mn+mnmfi.0 mm:.r aofi.0 m mnm.o nH0.o mfi0.o :0.m:mm.0 m0+mmmfi.0 mm.0 m0.mmmu.0 mam.¢ nmc.n m n:n.o mmo.0 mm0.0 30+mmmm.0 m0+mamx.0 cm.“ 30+moma.0 ¢or.u umc.0 m mnm.o 000.0 000.. m0.000m.0 m0.mmn:.0 «0.0 30+m0m«.0 :nfi.p “39.0 : m0m.o :m0.0 «00.0 :0.mm:m.0 m0.mmmo.0 mm.m 30+mumm.n mm:.u «00.0 m Hmm.c mH0.0 m00.0 m0+maaa.0 00+mn0m.0 mm.0 mo+mHmm.0 amfi.< m:n.n m Hwn.o mmo.o 0H0.n :o.m¢nm.n mn.mnma.o nfi.m mo.mmmn.n «mm.n rw¢.n m mfio.m 000.0 000.0 m0+mumm.0 00.mafim.0 “3.0 30+mxmm.0 n:H.0 :mc.0 s 0mm.m 000.0 :00.0 m0+mmaa.0 mn+mm:n.0 Hm.0 m0+m¢m_.0 mm".0 «00.0 n 0:m.m mma.0 mon.n m0+mmmo.0 a0+mn¢m.0 00.ma n0+m0::.0 mow.v 00n.0 m 003.: 000.0 mm0.c :0+mwna.0 m0+mafin.o 0H.m m0+mnm~.0 mm:.u r0—.0 m 000.m mom.0 mmm.c mo.mm:«.0 m0+mm0H.0 mm.~m m0+m~mm.0 anm.fl mmm.0 m mmn.m mmzmz am A>mzv HuA>mzv “Ammv Amv A>mzv Aq+ou.mmvm oxmfiqmv go ~n\qm Ammvau “.mmvm go am a xm .>m2 omumm .A.m.mvo¢mo .mumgmfimumm Hmcoflumunfl> m>fluomHHOUII.vI>H mqm¢H 84 000.0 000.0 0030030.0 30+0050.0 03.0 00.0000.0 000.0 000.0 5 530.0 000.0 000.0 00.0000.0 00+0m00.0 30.0 00.0000.0 050.0 03..0 0 035.0 000.0 000.0 00.0030.0 00+0000.0 50.0 00+mm0m.0 050.0 03 .0 0 050.0 300.0 000.0 00.0000.0 00.0000.0 00.0 00.0500.0 «00.0 mm0.0 m mmm.0 000.0 300.0 30.0005.0 00.0000.0 00.0 30.0000.0 000.0 aor.0 0 003.0 000.0 000.0 30.0005.0 00.0000.0 50.0 00.000«.0 000.0 050.0 3 000.0 000.0 500.0 00.0000.0 00+0030.0 50.0 50.00mm.0 «00.0 000.0 0 000.0 000.0 000.0 m0+mmm0.0 00+m500.0 mm.0 00+mmmm.0 300.0 m30.0 0 000.0 000.0 000.0 mo+mmm0.0 00+m000.0 00.0 00+mmmm.0 n00.0 030.0 0 00030 000.0 000.0 30.0030.0 00+m000.0 50.0 m03m000.0 500.0 c5r.0 3 000.5 000.0 000.0 3o+mm0m.0 003mmma.0 00.0 mo+mmmw.0 000.0 000.0 m m00.5 300.0 000.0 m0+mwm0.0 00+m03m.0 50.0 m0+mm30.0 «00.0 0m0.0 3 00m.5 000.0 000.0 m0+m0m0.0 00+momm.0 mm.0 00+m500.0 000.0 330.0 m 00m.5 000.0 500.0 00.0550.0 00+mm0m.0 05.0 00.0300.0 000.. «3r.0 3 303.5 000.0 300.0 m:+mmm0.0 00+mmem.0 00.0 00+m0m5.0 «00.0 3a0.0 0 000.5 030.0 000.0 30+mo0m.0 00+mn00.0 00.0 003mm3c.0 000.0 000.0 m 000.5 000.0 000.0 30.0300.0 00+05am.0 m0.0 00+m000.0 033.0 000.0 m mnm.o 000.0 000.0 30.0mmm.0 00.0330.0 00.0 30.0000.0 050.0 30r.0 0 535.0 000.0 500.0 30+m000.0 00+mm30.0 03.0 30+m000.0 n03.0 300.0 0 55m.o 000.0 000.0 00.0000.0 00+m003.0 00.0 30+0000.0 n00.0 030.0 3 000.0 300.0 330.0 30+mm00.0 00+m00m.0 30.0 30+M000.0 0¢3.0 000.0 0 030.0 000.0 000.0 30+m3mm.0 00+m500.0 05.0 00.0035.0 000.0 3mr.0 0 000.0 030.0 000.0 30.0mmm.0 00+mm00.0 00.3 50+mm00.0 303.0 c00.0 m 00o.m 500.0 300.0 00.0500.0 00+005o.0 00.0 30+0005.0 «30.0 o00.0 3 050.0 500.0 m00.0 m3+m0m0.0 ~0+mmm3.0 00.0 00+mmm0.0 000.0 “00.0 0 030.0 300.0 550.0 00.0000.0 m0.0000.0 00.00 50.0003.0 500.0 00n.0 0 503.3 500.0 000.r 30.0300.0 00+mnm5.0 00.0 0040000.0 (03.0 300.0 0 000.0 000.0 000.0 00.0000.0 00+0000.0 50.00 mo+mm5m.0 «00.0 500.0 0 005.0 mmzmz gm 0>mzv 0u0>mzv 00003 003 0>m23 3+0: .03 0 0:0: 0.08 .00 000300 303 no 0.03 m .00 .00 q xm .>mz mmumm .0.m.mvovmu .mumumfimmmm HmcoHumunfl> m>wuomHHOUII.mI>H mqmda 85 4.6 L=l States Figure 4.9 shows the experimental cross sections obtained for three £=1 states at 5.899, 6.944 and 8.270 MeV. The solid curves drawn against the data of 5.899 MeV are the results of collective model calculations. It is seen that the fits are very poor, therefore deformation para- meters were not obtained for £=l states. This is because that under the incompressibility constraint, the i=1 vibration corresponds to the oscillation of the center of mass of the nucleus, which of course is not the excitation observed. A microscopic description which accounts for the lst and 2nd 1. states is given in Chapter VI. 4.7 First Excited O+ State The collective model using only an R-vibration also failed to reproduce the shapes of the distributions for the first excited O+ state (3.35 MeV). Calculations for this state based on a generalized collective vibrational model have been carried out by Satchler (Sa 66a, Sa 67) but no data were available at the time those calculations were made. These calculations were redone by the author of this thesis. The potential U(V,R,a,r) was assumed to undergo oscillations of the form: U=wuowHHoo uoom ms» 30£m mw>nao naaom was .mmumum Hua mo mcowusnwuumfio amucmfiaummeIl.m.v musmwm so au 8 a... 38.3w .83 m 3.3 o ON. 00. 00 OO 06 ON ON. 8. 8 8 O? + ON ..O ON. 00. 8 OO O? ON .06 v A I 1 v .. r>ozn~ . L ,>.:n~ ..... . . >53 .. 1 fl ** A H O 6.. o # n H H 0 A H O L n 1 x u v . mo; " @_.O .1 . 1, _.o r 1 u + . . >028 . + . .. . 4 1 m I A .53... ........ ... 1 . >ozon n + + L w .. A I n + w. n . m T + + _.O n] o O._ n: _.0 >22 mm 1. . u .. . r r +* +¢ 1>Znn ....... . o r >023. . : + r . . v 1 o v H + M m . m I 1 v . n . ma :1: + L_.O V . . 0.. l .o is. z: z 1.. , . a. * 4 > 20‘ o 0. 6.6 .0. 1 >02 0? w * + W H o o r h . w m . m r Leo W . o; w L_d fl *+* L i o o o v .A H 2.3 8 W n 2;: 8 2;: 83 I M O? L n O' . . r p p » p p p L F F > A p r h p p p L p p p p b p 0.0. r p p p h r P h p b b _ k 4 O— (Jsxqw) vp/op 87 with the constraint: 2 2a 2 6R 2 2 2‘6a (3R +17 )——'+(R2 +TT a2)§%-+211 a —5' =0 where U(r)=-Vf(x), xiii, f(x)==EV(rRR>df‘X’39§ (5) all vibrations AU5(r)=AUl(r)+AU3(r) 88 2 2 2 2 2 2 B=3R +n a I C=3R +n a R +n a 2n a , D=B/C A computer program was written to generate the form factors corresponding to the potentials described above. These form factors were obtained using the prescription dis- cussed in Section 4.1.2. The parameters of the potentials are the same as those used in other calculations. Form factor decks adaptable to the JULIE code were part of the output of the program. Fig. 4.10 shows the results of these calculations at Ep=25 MeV and a comparison with the data at the same energy. The fit to the data is seen to be poor. Also note that the calculations are out of phase with the data. On the other hand, previous calculations by Satchler were reproduced indicating that the program is correct. Calculations were also done at 30, 35, and 40 MeV with similar results. Also calculated were the angular distributions in which the parameters V, R, and a were varied on a grid-like basis. In addition, complex form factors within the framework of the a-vibration theory were tried. No fit was found. Next an empirical form factor of the following form was assumed —(r-R+a)2 -(r-R-a)2 F(r)=A[e a -Be a ] After searching on R, a, and B it was found that the data could be fitted using the form factor shown in Fig. 4.11. 89 E 40 q . Ca(p.p’) E 825 MeV « ~ 01553.35 MeV * $- .. - é A '0 — as + . B 9 O i 9 + E E O 0 ~ 8 '+ l 3 ’ ‘ 9 I62 i 3 SATCHLER‘S THEORY . - BREATHING moo: - . v-cousr. —-— . R-cousr. ------ " a-wanmou' '02 9 '3 :\, z '1" ' ‘ Q - 4 .D .EiIO’? 3 ”<3 E 5 13 . J “O .. E F 3 b d '°°5' ': r w 'O.' l l l l l L J j l l l L l 1 o 20 40 6o 80 I00 I20 I40 Gama) Figure 4.lO.--Results of generalized collective model calculations based on Satchler's theory. 90 .mufim .m.m HMUAHAQEm can wumum o umnfim may mo coausnauumwo Hmucmfiwuwmxmll.aa.v musmwm + 3.230 E ._ .ow..ow..om.ow.mw.ow.o o_ao~oncn~_o w o. .9 n. .u M T . mxw. n u .w: m m r q I I. . .4 m l. . T“ r ...I To. I n H n 2+9 “ W m m m I abuklfll I“ I. .u ......1 432;“. , ..o. ... ow... .m . . 82.8 5 Sun .I / . ... co... .m r r . I“ a H n >22 magma wmo r8210 3.28.3.0 mo“. n w . co m ".9..an EEO“. Jwlu :mmm Ref. 5 (Ya 64); Ref. 4 (Gr 65); Ref. 7 (Sp 65); Ref. 8 (Li 67); Ref. 10 (Bl 63); Ref. 11 (Bi 69); Ref. 12 (Br 66); Ref. 13 (Se 67); Ref. 14 (Fu 69); Ref. 15 (Le 66); Ref. 16 (Le 67); Ref. 17 (Li 68); Ref. 18 (Do 68); Ref. 19 (Po 69); Ref. 20 (An 69); Ref. 21 (Ma 68); Ref. 24 (Me 68). 96 5.2.1 States Below 6.585 MeV The spins and parities of low-lying levels below 6.585 MeV have been well determined. The most complete set of spin assignments was given by Anderson e£_al, (An 69) who have done extremely precise (p,p'y) measurements. A summary of the previous assignments has been discussed by Seth g£_gl. (Se 67). The spins and parities of 3.3so, and 69% of $1 (KUO), in which [(3.3/2 f7/2] is the largest component (Ku 66). Hence the theoretical configur— ation proposed for the second 2- state are deformed plus -1 . . . the [d3/2 f7/2]. The predicted energies for the first and second 2' states are 6.4 and 6.85 MeV which closely agree with the experimental values of 6.02 and 6.75 MeV if the 99 latter is assigned to 2-. The agreement between theory, (d,n) and this (p,p') experiment suggests that the 2- assign- ment is favored. It has been pointed out by Seth gt_31. that appreciable L=3 transition can be mixed with Lp=l with- out affecting the quality of Lp=l fits but the inverse is just the opposite. It was also found that in their data the smallest angles of observation for this 6.75 MeV state stopped at about 30°, where the maxima of Lp=3 distributions occur. Their pr0posed Lp=1 peaking at about 10°, without comparison with data, may result in their overestimating the Gz=1 strength. The 6.91-6.93-6.95 MeV triplet: individual angular distributions for 6.91 and 6.95 MeV states were obtained and their Jw-values are positively assigned as 2... and 1- (see Fig. 4.4 and Fig. 4.9). Metzger, using y-resonance techniques (Me 68) has concentrated his effort on this 6.91- 6.93-6.95 MeV triplet, and he was able to identify the first and third members as 2+ and l—, in agreement with this (p,p') finding. The 6.95 MeV level has also been assigned 1- by proton stripping reactions. As has been mentioned in Section 3.9, where the analysis of this triplet was discussed in detail, the middle level of 6.93 MeV may be a high Spin state(:3). The 7.114 level: The spin and parity of this state has been tentatively assigned (3)- by many authors of previous works. This assignment was first given by Gray EE_Eln in their (p,p') experiment using relatively low proton beam energy. This level was also observed by the 100 (0,0') reaction (Li 67) although no Spin identification 3.d) was made. An Lp=l transition observed for this state in (He (Br 66, Seth 67, F0 70) and (d,n) (Fu 69) reactions leads to the (3)- assignments by these authors.) A contradictory result was found in this (p,p') experiment. The angular distributions of this state resemble those having L=5 transfer and are very similar to those of the 5.62 MeV state (see Fig. 4.7). At Ep=25 MeV, the distributions of these two levels agree very well with the L=5 collective model prediction. At Ep=40 MeV, the distributions are intermediate between L=5 and L=4. In any case, the angular distributions of the 5.62 and 7.11 MeV states are remote from those of L=3 transfer. Other evidences of similarity between these two levels can be seen from the 39K(p,y) experiment performed by Lindeman g£_al..(Li 68). The gamma-ray branchings of both the 5.62 and 7.11 MeV levels were found about the same, namely 70% to 3.74 MeV level and 30% to 4.49 MeV level. The forementioned calculations by Gerace and Green suggest that the second 4- state is essentially a collective state with over 80% of 5E:IH strength. The predicted energy is 7.65 MeV. Thus it is believed that the 7.11 MeV level corresponds to the second 4- of Gerace and Green's scheme. The Lp=l (in stripping) transition property of this state cannot, perhaps, be interpreted by the simple particle-hole picture of the shell model. 101 It appeared that the present data on the 7.11 MeV state are most consistent with a 4- assignment. The 7.53 MeV level: This state has been observed in (He3,d) and (d,n) experiments and tentatively assigned as (2)-,based on the shell model. In the present experi- ment, this and the 7.56 MeV levels are not separated and analyzed by decomposition method (Section 3.8). The 7.53 MeV level is found to be excited by a L=3 transfer in agreement with the results of proton stripping reactions. The 7.56 MeV state is identified as 4+. The 7.57 MeV level observed in the (a,a') experiment (Li 67) may correspond to this state. 5.2.3 T=l Analog States 40 The T=l analog states of K have been assigned by Erskine at 7.660(4'), 7.696(3‘), 8.465(2') and 8.553(5-) MeV. His proposal was based on the results of his (He3,d) data and on Enge's (d,p) experiments (Er 66, En 59) and of the observation of the lowest T=l state in Ca40 by Rickey et al. (Ri 65, Ka 68). Also these experimental results agree with the calculated excitation for the lowest K4O analog 40 1 states in Ca f7/2] configuration. Experimentally, in [d3/2 this has been further investigated by Seth et al. and Fuchs et al. Both groups have confirmed Erskine's identification. Fuchs et al. even extended this technique to identify the -l T=l, [d3/2 p3/2] quartet. 102 In the present experiment, the 7.660, 7.676, 7.696 MeV triplet was not resolved and the Jfl—values of the 7.660 and 7.696 MeV states are taken from the results of authors mentioned above. The 8.424 and 8.535 MeV levels are observed to be L=3 and 5 transitions respectively, consistent with results of the stripping reaction experiments. The 1P3/2] T=1 quartet was proposed by Fuchs et al. to [d3/5 consist of the 10.040(o‘), 9.435(1‘), 9.408(2') and 9.404(3‘) MeV levels. At Ep=35 MeV, a level at 10.045 MeV was seen having L=5 transfer. It is suspected that this may not be the same level observed by Fuchs gt_§1. No angular distribution for the 9.435 MeV state was obtained here. A doublet at 9.411 MeV with an L=3 transfer angular distribution was observed which may corre3pond to the 2- and 3- levels at 9.408 and 9.404 MeV. 5.2.4: States Between'7.6Tand8.89MeV Aside from the T=l analog states discussed in the last section, there are a few even-parity states which lie in this region. The 7.867, 8.092, 8.578 and 8.743 MeV levels were identified as 2+ and the 7.928, 8.371 MeV levels as 4+, in agreement with the results of (a,a') experiments. It is interesting to observe from the Table V-l that the (a,a') experiments missed all the T=l states as expected from the selection rule AT=0 for the inelastic scattering of Alpha particles. There are two L=l states observed in this region. The 8.271 MeV level (see Fig. 4.9) is tentatively assigned 103 (0,1)-. The possibility of 0- was assumed since the angular distributions of unnatural parity states are not distinguish- able from those of natural parity states with spin just one unit higher. The 0- component of the T=Oi[d3/;lp3/2] quartet was tentatively assigned by Fuchs g£_31, to be one of the 8.271 or 8.371 or 8.931 MeV levels. In the present experiment the 8.93 MeV level was very weakly excited (about 30110 ub/sr at 300 and 8:4 ub/sr at 60° at Ep=25 MeV) and no angular distri- bution could be obtained. The 8.371 MeV level has been identified as 4+ in this and two (a,a') experiments. As in the case of the 7.114 MeV level, the nature of LP=1 transition observed in (d,n) reaction for the 8.371 MeV state is open to further investigation. Another L=l state is the 8.664 level which was also weakly excited in this experiment. The 8.113 MeV level is assumed to be (2,3)_. 5.2.5 The High L Transfer States and Levels Above 9 MeV Several States having spins possibly equal to 6 or greater were observed. The characteristics associated with high L transfer in the (p,p') reaction is that the angular distributions of such excited states are isotropic at low proton beam energy,as can be seen in Fig. 4.8. The 8.186 and 8.974 MeV levels are observed with L=6 transfer, and their Jfl-values are tentatively assigned as 6+. The L=7 transfer to the 8.848 MeV state and its angular distributions show systematic agreement with the collective model predic- tions at four beam energies. This state is tentatively 104 assigned J"=(6-,7-). The same assignment is given to the 9.237 MeV level but with less confidence, for there is only one angular distribution analyzed and compared with theory. It is believed that these high spin states in the Ca40 nucleus have been observed and identified in this experiment for the first time. Extreme care has been taken in the analysis of these states. However, further investiga- tions by other types of reactions are needed to confirm these findings. Finally, the spins and parities of a few levels above 9 MeV excitation energy are tentatively assigned from their apparent L-transfers, as listed in Table V-l. Due to the fact that the density of states is very high above 9 MeV (see Fig. 5.1), one to one correspondences with the results of Fuchs et al. and Leenhouts et al. was not made. 5.3 Comparisons of 6L's and G's Table V-2 summarizes the experimental nuclear deform- ations, 6L, we obtained, and includes the present and previously reported experiments. Only comparable results are listed here. It can be seen that for the 3.73 MeV(3-) state the energy dependence of 6L on the incident proton energy is not obvious. For six beam energies and three independent experiments, the deformation was found to be more or less a constant 1.4 fm. The observations of two (a,a') measure- ments are consistent with each other and incidentally very .LUb momma mam? ml”:— xme amt on .m6m Ase eat m .M6m .me mmv e .662 new 62v m .M6m Ame moo e .mmm me.o na.o sa.o m mnm.m Hm.o m~.o ma.e m mmm.m ~m.o e~.o ~m.o mm.o e Hem.m oe.o em.o m~.o em.o m~.o e ems.» ~m.o m~.o m~.o N nem.e «we.o oe.o n~.o m see.e em.o Hm.o oe.o He.o mm.o m mmm.e mm.o me.o oe.o ~m.o ee.o oe.o m mm~.e oe.o mm.o mm.o me.o Hm.o om.o m mme.s me.o oe.o em.o ee.o «v.0 me.o m eom.m em.o em.o mm.o mm.a ee.H oe.a mm.e m eme.m >62 >62 Hm >62 om >62 mm >62 ea >62 mm, >62 ow A>62V ommuoma 2 OH .662 m .w6m e .662 m .M6m e .M6m .mxm ness 2 m 2.6.6v i.e.sv i.e.sv i.e.mv i.e.mv 6A.m.mv q @ .mCOHumEuommo unmaosz wo coweummEOUtu.mn> mqmde 106 close to the (e,e') result, but only 2/3 of those obtained from (p,p'). The deformation of 3.90 MeV(2+) state is independent of proton energy as well as of the type of scattering particle. Other even-parity states show about the same trend. For the 4.49 MeV(5-) state, the (p,p') deformation is again about twice as large as the (a,a') findings. The results for the 6.28 MeV states are quite consistent in every case. The qualitative agreement between (a,a') and (p,p') experiments on 6.58 state can also be noted, except for the 17 MeV (p,p') work. Statistically one finds that the deformations extracted at higher energy are consistently smaller than those at lower energies in both (p,p') and ( , ') experiments. This trend of energy dependence may result from the model and analysis procedure used. A comparison of the reduced transition probabilities with (e,e') and y-decay experiments is made in Table V-3. The entries for the present experiments were from the calcu- lations described in Chapter IV. Only those transitions with 100% to ground state branching, i.e., B(EL;O+L) are compared. As can be seen from.the table the G values obtained in this experiment agree very well with the majority of all other results, especially those of Eisenstein‘22431. (Ei 69). It has been pointed out by these authors that their findings are relatively parameter-or model-dependent. .Amm HOV paw H.v cowuomm mom Oman om cues coausnwuumfip mmumno EHOMAGS eucmam>wsvw flanmmo enmm.au .Amo 62v Hm .e6m .xme see om .emm .Ame one as .e6m Ame 62V eN .M6m .Amm one we .emm .Ame amt ea .M6m .xee ems He .u6m 107 e me.one.e ne.a m.one.~ e.e m~.onm~.~ s +hsem.ev+m mo.enm~.o me.en~.o ee.en-.e mo.onme.o s +2e~e.mc+m H.H me.onme.o mo.ene~.e . +Aeem.mv+m e.e m.ene.e ~m.enee.m 6m.ens ms.ene.~ e.~ ~.onmo.~ = +Aeee.mv+m e.mne.e~ m.one.e n.2m e.~ne.mm A.n.mv+e Asme.mv-m Hm .66m om .u6m me .e6m em .u6m OH .662 He .e6m >62 es “sees: no noses n ucmmmHm A».e.sv n>.n.nc I».n.nc see .662 ..6.6v i.6.6v 6i.n.nc consensnne .muwca OHUflunmm mamcem xmoxmmflmz 2H Ammvw .mwflueaflbmnoum coauwmcmua pmuspom mo mGOmHHmmEounu.ml> mqmde 108 The striking agreement between this (p,p') and Eisenstein's (e,e') experiments indicate that the prescription proposed by Gruhn e£_gl, (Gr 69), for the (p,p')B(EL)'s calculation allows the (p,p') results to be scaled against the (e,e') data reliably. A comparison of B(pp'; 0+L) and B(aa'; 0+L) is shown in Table V-4. TABLE V-4.--Comparisons of Reduced Transition Probabilities Between (p,p') and (a,d'). L Ex L (p,p') (a.a') this work (Li 67)* 3.90 2 2.05:0.20 2.910.5 5.62 2 0.1310.05 0.710.2 7.87 2 0.9210.15 . l.8:0.4 8.10 2 0.38i0.06 2.1:0.3 3.73 3 28.7 13.0 23.613.5 6.29 3.1 10.3 6.6:l.0 6.58 2.5 10.3 3.8:0.6 7.94 2.2 i0.2 5.610.8 8.38 2.0 $0.2 4.310.6 4.48 20.6 i2.1 17.712.7 *Also A. Bernstein, in Advances in Nuclear Physics, edited by M. Baranger and E. Vogt (Plenum Press, Inc. New York), CHAPTER VI MICROSCOPIC DESCRIPTION A great deal of work, both theoretical and experimental, has been directed towards the understanding of the energy 40 level scheme and transition rates in Ca in terms of the shell-model and its extensions (Gi 64, 67; Ku 66; Ge 68; Le 67; Di 68; etc.). The properties of the negative parity 40 have been most vigorously investigated. The - states in Ca RPA seems to give a reasonably good description of the salient features of these states which are formed predominately, although not entirely, from single particle-hole excitations. Positive-parity states are likely to contain large admixtures of many particle-many hole excitations, i.e. deformed com- ponents, and are not so easily described. Recently some progress has been made in describing the (p,p') reaction in terms of a direct interaction between the projectile and target nucleons through an effective force. The properties of the effective force are largely dictated by the empirical two-nucleon potential. In partic- ular, it has been shown by direct calculation that the bound state reaction matrix ("bare" effective force between bound nucleons) is a good guess at the "bare effective force in 109 110 the inelastic scattering process when the laboratory energy of the projectile is in the range from 15-70 MeV (Am 67; Sc 69; Pe 69; etc.). This is based on the studies of strong, normal parity inelastic transition and the real well of the optical potential which mainly test the strong central, iso-scalar component of the force. In these studies it was found that exchange effects are important, as was originally pointed out by Amos, Madsen, and collaborators. In the present work, microscopic distorted wave approximation calculations are performed for some of the 40 negative parity states of Ca and comparisons made with our (p,p') data. Random-phase-approximation state vectors of T. T. S. Kuo (Ku 66a) are used for the states of 40Ca in the calculation and exchange effects are included approxi- mately in the DW calculations. The Kallio-Kolltveit (K-K) force and the central part of the Hamada-Johnston (H-J) force are used for the projectile-target interaction. The latter is basically the same force which has been used in the RPA calculation. 6.1 Theory The antisymmetrized distorted-wave transition amplitude for the spin—dependent nucleon-nucleus scattering reaction is given by 111 _ + TDW — p§rx (-) (+) , _ (+0 where t(0,1) denotes the particle-target interaction, the x's are the distorted waves mentioned in Sec. 4.1 + O O I O a ,a are shell-model state creation or annihilation p r operators, |A>,|B> refers to the nuclear states, and |¢> is the single-particle shell-model state. The first term in TDW is the usual direct matrix element, the second is the exchange term which arises auto- matically from the antisymmetrization. Details concerning both the effective interaction and the nuclear wave functions used in this thesis are discussed in the following sub- sections. The procedures used to reduce the transition amplitude to partial matrix elements using the nucleon- nucleon interaction t(01) = 2 (-1)*+Yt (E st st AY s s T T 01)OA(O)O-A(1)TY(O)TY(1) have been given in detail by McManus and Petrovich (Pa 70). This program permits the separation of the details of the interaction model and the nuclear structure from the dis- torted wave calculation. With the approximate antisymmet- rization the nuclear wave function and the radial form of 112 the interaction are further separated. The main features of their formulation are: l) 2) ~ The form factor F LSJ,T LSJ'T is related to the transition density F by the expression ~ FLSJ,T _ .L 2 . LSJ,T (r0) - i frldr v (r0,r1)F (r 1 STL l) LSJ,T The transition density F (r1) contains all of the information on the nuclear wave functions and their coupl- ing scheme. The function vSTL represents the radial form of the interaction including the exchange effects. Its explicit expression is 6(r -r ) . _ (l) 2 0 l VSTL(r0’rl) ‘ [tST(r01) + A (40)] r2 0 where the first term is related to the direct inelastic scattering and the second is the exchange. The quantity A(l)(Ag) is the first term of the Taylor expansion of the Fourier transformation of th; they are related by E _ g S'T' tST ‘ S'T' AST tST -'X-E 2 _ 1 01 E 3 A(x ) — je tST(rOl)d r01 _ (1) 2 2_ 2 dA — A (A0) + (A Ao)——71 + . . . d A2=A2 113 This is used to reduce the form of the exchange component of the partial matrix element to that of the direct component, so that the above expression for F can be achieved. The simplest approximation for treating the exchange component is done by making A(AZ)+A(1’. where A3 is defined as 2__2 _ 2 )‘0_kLAB- 21leLABm ' ‘1’(x hence A g) is energy dependent. This approximation is treated for the K-K and other effective range forces. 6.1.1 Effective Interaction In analogy to the shell-model calculations using the G-matrix, the two-body interaction to be used in nucleon- nucleus scattering calculations is given by _ _ Q = t — v v 5:3; t, t¢ v? where v is the "bare" nucleon-nucleon potential, and Q is the Pauli operator which excludes all the occupied states. The energy denominator e is defined in accord with the conventions of Kuo and Brown (Kuo 66), but is appropriate for the scattering problem (see the appendix of (Sa 67)). The effective interaction used in this work was assumed to be 114 .. _ z . Veff _ i t(l'a) where the summation is over all active nucleons. It was also assumed that the two-body interaction t is 12331, state independent, and a scalar separately in spin, isospin and coordinate space. It has the form given by Kerman, McManus, and Thaler (Ke 59), i.e., .+ . + t(i,a) a) + V10(ria)0i.0a = too‘ri + t ( )+ + + t ( )+ 01 ria Ti Ta 11 ria Oi oaTi Ta In the above relations the double subscript on t is to be read as ST, referring to the multipole components of the force in spin and isospin space respectively. Approximations which under certain conditions give simplified expressions for t(ia) have been given. This can be written, if the imaginary part of t is small, as t ~ tB - intBQG(e)tB where tB is real and satisfies the relationship — _ 9 Using the Scott-Moskowski separation method and taking the Hamada-Johnston potential (Ha 62) for the nucleon-nucleon potential v, Kuo and Brown have shown that the attractive even component of t can be represented by B 115 H < (attractive, even) where T denotes the tensor component of the long range part of vg of the H-J potential. It has been shown that (Ku 67, Pe 70) if the average effect of the odd state interaction is small, then near the target Even _ E . E E t — tB intB Q VT£ + -_?E;—_' (triplet, even) = VEE (singlet, even) where 512 is the tensor operator. The two—body interaction t used in the present calculations was further simplified such that l) the imaginary part of t was neglected, 2) the tensor part of potential was not included. One might expect that the calculations so performed for certain states would be quite inadequate for comparison with the corresponding experimental results. They were still used, partly because the existing program was not in readiness to include the tensor and imaginary parts of t 116 and partly because the effects of neglecting the tensor force was one aSpect of the study. The types of interactions under investigation are discussed briefly in the following: A. The Kallio-Kolltveit force: In using this force t was approximated by TE SE t I V (V ) The S-state condition was relaxed and the interaction was allowed to act only on even states. It was pointed out by Petrovich, McManus, and collaborators (Pe 69) that this requirement may lead to errors of up to 20% in the strength of the interaction. The explicit form of the interaction (Ka 64) is the following: ngs(r) = + w ’ r:0.4F = _AT’Se-GT'S(r-O.4) r>0.4F where AT = 475.0 MeV, 6T = 2.5214E"1 AS = 330.8 MeV, as = 2.4021E'l The above parameters were determined by fitting the potential to the scattering length and the binding energy of the deuteron. 117 B. The Kuo-Brown Force: given by ref. (Ha 62, Ku 67). The t was approximated by taking only the central force I C. Yukawa Force: A real 1F range Yukawa "equivalent" to the K-K force was used V(r) = V mr, m = 1.0F. e-mr/ O The strength V was determined by a normalization procedure 0 similar to that used for deformation parameters (see Sec. 4.3). 6.1.2 Wave Functions Extensive calculations for the ground state and the 40 in terms of particle-hole configur- odd parity states of Ca ations using the RPA method, have been carried out by Gillet and Sanderson (Gi 64, 67), Kuo (Ku 66), Leenhouts (Le 67), Dieperink gt_al. (Di 68) and Perez (Pe 69a). Effects of spherical and deformed state mixing between the odd parity states have also been reported by Gerace and Green (Ge 68). Moreover the simple shell-model picture for this nucleus was given by Erskine (Br 66), Seth et a1. (Se 67) and Fuchs et a1. (Fu 69). 118 Gillet and Sanderson predict the results of the diagonalization of the matrix elements of the effective two- body force taken between the single particle-hole shell-model states. The unperturbed energy of a particle-hole configura- tion is the apprOpriate value determined by experiments. The energies for proton particle-hole states are taken from 41 39 . -1 those of Sc and K With AE(d3/2f7/2) equal to 6.71 MeV, whereas for neutron states are from Ca41 and Ca39 with u '1 = ' ' l ' AE (d3/2f7/2) 7.37 MeV. The difference in AB and AB is accounted for by the average Coulomb energy shift. The effective force parameter of the spin and iso-spin dependent Gaussian potential (Calcium force) is 40~45 MeV and the oscillator parameter is 0.53. Isospin was not considered a good quantum number thus their results showed strong T mixing. States with calculated level energies below 10 MeV are shown in Fig. 6.1 along with the results of other inves- tigations. However, Seth e£_§l, and Fuchs et al. found from their proton stripping experiments that the odd parity excited states of Ca40 can be explained rather well by a predominantly simple shell-model and that T-mixing of low- lying states was much less than predicted. A summary of "configuration, spin and isospin" assignments to the Ca40 negative parity states in terms of [d;}2f7/2] and [d;}2p3/2] shell-model states has been given by Fuchs e£_21, In his pure RPA treatment of the odd parity spectrum of Ca40, Kuo used a G—matrix derived from the H-J potential 119 eeq._.qqq....__e__._e....q22._...e_eej4_1j+e_.1+ee_.ej_14_...e.._...eees — (4,5Y ——lmfl‘ -——(2m' 3.. l O —O —O —2'l 0 .u-v.n ....- l 044 _:___ >__»_»___»L..e___._e_e___.___.e...__#.e__._e__.r>_»_t.__.e_.__t._.__...t._ I '0 .0! 'nl' I. 312 r.) r J3 UH} wnu m2 3.5 4..n, .. ._ . Z3 4.2 3 a mu 5 34 ll OIIOOO O .....11... o...- 234343U| 2 OIIO 00 I .I-u—I_O -I—0 -252 03 := __ — 2:0 — 3:0 — 270 30.0. 3322 .__4- —.3' .6 .o..o .o. 3 2| 4 ... 2|4 00 .... I 4 .o .o .o .o 3 2 4 5 —4 _ . 3 5 ——'SK) -— 570 3K) :0 -—370 —5 — 370 O l —3 Av 03 00 7! A>o_2v ab. R4 335 cozozuxm 4. 29 GERACE GREEN EXPERIMENT PEREZ DIEPERINK of al. T.T.S. KUO G ILLET SANDERSON --Theoretical and experimental energy levels for negative parity states of 40Ca. Figure 6.1. 120 for diagonalization. His Spectrum is shown in the second column of Fig. 6.1 for the comparison with Gillet's results. Both RPA calculations encountered the difficulty of putting too much strength into the octupole transition to the ground state from the first 3- state. Dieperink's calculations used the modified surface delta interaction (MSDI) in both the RPA and TDA formulations, using a [d;}2f7/2] Splitting of 7.3 MeV. Their diagonalized wave functions are very close to the unperturbed particle- hole states. The positions of the first four T=l states were successfully predicted. Gerace and Green have constructed a model of mixing shell-model 1p-lh states with 3p-3h deformed states to describe the odd parity states of Ca4o. Their procedure was to start with RPA wave functions which were obtained using AE(d;}2f7/2)=5.4 MeV and SPE II. Kuo's particle-hole matrix elements were used and the effects of core polarization were included. The 3p-3h deformed states were constructed by first coupling Nilsson orbits no. 14 and no. 11 (Ge 67) to obtain a lp-lh K=l wave function, then recoupling this to a 2p-2h wave function to get the 3p-3h wave function. Finally matrix elements of the H-J potential taken between the J deformed states were obtained and the diagonalization was carried out. The diagonalized wave functions contain RPA wavefunctions and deformed l3p-3h> wave functions as illustrated in their paper. Their calculated 121 Spectrum is in good agreement with the experimental one below 8 MeV. Fuchs e£_al, have derived the Spectroscopic factors for their (d,n) work using Gerace and Green's wave functions and assuming the K39 ground state to be a pureid3/2 hole. They found that the theory agreed with experiment very well except a few discrepancies. Goode (Go 70) has calculated several E2 decays for the low-lying T=O odd parity states of Ca40. It was shown that a pure RPA description of these decays was not satisfactory, whereas Gerace and Green's picture provides a consistent explanation of the B(E2) values. In the comparison with the results of this thesis, several predictions of Gerace and Green were supported. For example, the deformed nature of the first 1- state at 5.90 MeV and the predicted existence of the level sequence 3-, 2-, 4- around 7 MeV are confirmed. . The purpose of this section is to summarize some of the current theoretical descriptions for the wave functions of the odd parity states of Ca40, so that one can estimate the uncertainties in the DWBA calculations due to the wave functions used. In this thesis T. T. S. Kuo's wave functions were used. It seems to be clear that Gerace and Green's wave function should have been used, but the existing program did not include the code to treat their wave functions. 122 6.2 The Calculations The procedure used in the distorted wave calculation for the microscopic model is the same as that for the collective model as described in Section 4.3. An external real form factor is input into JULIE. The optical model parameters used are the same as those for collective model studies. To obtain the external form factor, two separate programs NUCFAC and FBART were used. Both were written by F. Petrovich. 6.2.1 Transition Density NUCFAC calculates the transition density F LSJ,T which uses nuclear wave functions as input. When harmonic oscillator wave functions are used, F LSJ can always be written in the following form FLSJ(r) where N a Nb 0. 0.498 F- LSJ N+3 N -d2r2 N a r e I (2 +£)min (£+£'+2n+2n'-4) max 1 (in this work) The explicit expressions for the oscillator wave functions and for the transition density can be found in Petrovich's thesis. The wave functions used were given by Kuo (Ku 66a). ,Fcu the convenience of future reference, they are listed 123 in Table VI-l. The resulting transition density functions are given in Table VI-2. 6.2.2 Form Factors The form factor FLSJ’T was calculated using program FBART which performed the integration over the integrand V(r0,rl)F(rl). As mentioned in Section 6.1, v consists of two terms, a direct and an exchange. Form factor outputs can be obtained for direct only, or exchange only, or total. For the K-K force, the separation distances were ds=l.025F, and dT=O.925F respectively. The Fourier trans- (1) (A formation amplitude A g) is given in Table VI—3. For the K-B force, the separation distances were ds=dT=l.025F. The Fourier transformation amplitude A(l)(lg) is shown in Table VI-4. 6.3 Results and Discussions Calculations for the lst 1-, T=0 state; lst 2-, T=O,l states; lst, 2nd and 3rd 3-, T=O states; lst 3-, T=l state; lst 4-, T=0,l states; lst 5_, T=0,l states; and 6—, T=0,1 states were performed. The lst 3-, T=0; and lst 5-, T=0 states have also been investigated by Schaeffer and Petrovich. Comparison with the results of these authors and discussion on the calculations in this thesis will be pre- sented in the following subsections. 7.. 000.2 000.22 2.-0 000.2 000.22 0.-0 020.0- 000.0 002.0- 020.0 2.-0 022.0 000.2- 002.0- 020.0 0.-0 020.0 020.0 020.0 000.0 002.0 020.0 220.0 2.-0 202.0 000.0- 020.0 000.0 000.0 002.0 000.0 0.-v 220.0 200.0 000.0 000.0 002.0 020.0- 000.0- 002.0 200.0 000.0 2.-0 000.0 020.0 000.0 020.0- 020.0 000.0- 200.0 000.0- 020.0 0 200.0 200.0- 000.0- 200.0 020.0- 002.0- 002.0- 020.0 000.0 x 022.0 000.0 020.0- 200.0 020.0 000.0- 000.0- 200.0- 200.0- .220.0- 0 002.0 020.0- 022.0 020.0 020.0- 002.0- 000.0 000.0- 020.0 x 000.0 000.0 002.0- 022.0 002.0 002.0- 000.0- 222.0- 002.0- 202.0- 2 002.0 002.0- 002.0 002.0 022.0- 022.0- 000.0- 000.0- 000.0- x 020.0 0.-0 002.0 020.0 020.0- 000.0- 200.0 200.0- 200.0- 000.0 222.0 000.0 200.0 2.-2 002.0 020.0 000.0 000.0 200.0 200.0- 000.0 000.0- 000.0 022.0 000.0 0.-2 000 0 200.0 000.0- 000.0- 000.0- 020.0 000.0- 000.0 000.0 0.-2 020.0 220.0 020.0- 000.0- 020.0 000.0 000.0 002.0- 000.0 0.-2 002.0 200.0 000.0 200.0 2.-0 000.0- 200.0- 020.2 002.0 0.-0 2mms2 2MH02 2Am02 2am62 2AH02 2mm02 2%m02 2MH02 2Mm02 2mm62 2MH02 2mm62 2>622022 0._0 2x202 2\2n2 2x222 2\022 2\022 2x022 2\0n2 2\0m2 2x002 2\022 2\022 2\022 . ”a: . . .551! 0E3” 431:. ...nuu...n..‘,l.1wl.x I.” 2 n. 2... 1..-i. .. 1 .I'. |.I..L a ...E - . . . . . .. . .- ..g ...:P‘l.afi.1 1.1,Iv ‘. . . ll. . fila'Iu-flPEI- Idl- ..l11n11.l..l. f..... H :71.“ I.- n H .H. “I" l1:.l-...fl1.fll: -.:-|- . .... I.. .I1 .2 u .. . .. ll“. .11.) . .2002 .0 .0 .0 22 c6>2oc 0600 006226000 6>sz <20-.2-2> u22<0 TABLE VI-2.--Transition Density Function 125 LSJ(r) = NbaN+3rNe 3 2 CLSJ Eth N J ,T (MeV) LSJ,T =1 =3 =5 0’,0 7.144 110,0 -1.140 1.841 00.737 o',1 9.052 110,1 0.484 1.159 -0.527 1‘,0 7.767 101,0 -0.023 -1.373 0.484 1',1 8.449 101,1 0.053 0.146 -0.055 2',o 6.393 112,0 0.251 -0.441 -o.103 2‘,1 7.672 112,1 0.586 -0.714 -0.065 3‘,o 3.826 303,0 -l.128 0.909 313,0 -0.231 0.264 3‘,0 6.558 303,0 -1.180 0.512 313,0 -0.625 0.175 3‘,0 7.118 303,0 -0.457 0.114 313,0 -o.393 0.153 3‘,1 6.567 303,1 0.198 -o.003 4',o 5.407 314,0 0.051 0.082 4',1 16.521 314,1 0.104 0.072 5‘,0 4.323 505,0 -0.265 515,0 —0.175 5‘,1 7.610 505,1 0.178 515,1 0.204 6‘,0 0.316 6—,1 0.316 TABLE VI-3.-—A(l)(lg) for the K-K force. 126 Ep (MeV) A A oo 10 01 11 25 -206.0 40.0 _ 97.0 68.0 30 -180.0 33.9 26.3 60.1 35 —159.0 28.8 77.7 53.0 40 -138.0 23.7 68.2 45.9 (1) 2 TABLE VI-4.--A (10) for the K-B force. Ep (MeV) A00 A10 A01 A11 25 -166.5 31.5 79.5 55.5 30 -l46.0 26.6 67.7 48.6 35 -127.5 22.6 62.5 42.5 40 . -112.0 19.1 55.5 97.3 127 6.3.1 The 1‘, T=0 State: The major p-h components of the wave functions for . - -1 -1 this RPA lst 1 state are [ZPB/ZdS/ZJ’ [2p3/2d3/2] and -], . [fS/zdS/ZJ‘ The calculated angular distributions at 40 and 25 MeV are best fitted by the distributions of the 2nd experimental 1- (6.944 MeV) state. Figure 6.2 Shows good agreement both in shape and magnitude between theory and experiment if so assigned. The magnitude of the distribution F...- A at 5.900 MeV is about 10 times smaller than that theoretically predicted. Gerace and Green's (GG) calculations show that the lst 1- state is strongly deformed whereas the 2nd 1. is a very pure RPA lst 1. state. Thus the assignment of the RPA lst 1- state to the 2nd experimental 1- state is supported by Gerace and Green's theory. In other words, the micro- scopic DW calculations, the angular distributions obtained in this experiment, are in agreement with Gerace and Green's deformed model. 6.3.2 The lst 3-, T=0 State: The antisymmetrized DW calculations for the lowest 3- state have been previously reported by Petrovich and McManus (Pe 69), and Schaeffer (Sc 69). The results of the present calculations are shown in Fig. 6.3. For this 3- state the magnitudes and positions of the maxima are well reproduced at each beam energy. The overall shapes of the «experimental distributions are also qualitatively fitted .indicating that the energy dependence of the exchange effects 128 :- 4°CO (p'p’) E r7??? MeV—j : = 6.944 = - : g 8" MW TISKUO'S WE: ' A —— K.K.+ EX ‘ ¢ :1: : Z \ - .. n r- 2 E : j d - .. 'O ‘\ b .I - -= 1, Z : <2 : [ I b .¢¢. ‘ 4 O I. E— ".2. E 25 MeV : | 1 l L l l l l L l l l l ’ o 20 4o 60 80 I00 120 ecuweg) Figure 6.2.--Microscopic DW calculations for the 1-, T=0 state. 129 40 I I51 3-, T=O ......KK FORCE Co (0. p ) Em= 3.826Mev __ EXCHANGE; E, = 3,732 MeV (7.7.5. KUO) —— KK+EX ---- YUKAWMIF) " 5,, = 40 MeV F ‘ Ep 3 35 MeV --4 q T I fiTIT' .\ \ 1 / \ \. o “ \\ _ '2 :' \\ \. . ‘ : Z \ \\.\9. - L '\2::: F 1 1 T U UY—VVII \\q\- 1. § L 1 )- r- -( 1. .. .- 0 20 40 60 80 IOO 0 20 4O 60 80 '00 60" (deg ) MSU CYC Figure 6.3.--Microscopic DW calculations for the lst 3-, T=O state. 130 has been correctly accounted for. It can be seen from Fig. 6.3 that the contributions from exchange become increasingly important at the lower energy. The calculations using 1 fm range "KK equivalent" Yukawa force are also illustrated in Fig. 6.3. The distribu- tions are very Similar to those obtained by using KK+EX forces. The results of KB+EX forces are not shown because the shapes of the calculated distributions (in direct, exchange and total)were found identical to those using KK+EX forces, except that the predicted magnitudes were found to be about 25% lower. This similarity applies to the calculations for the 2nd 3’ (6.285 MeV) and the 5' (4.487 MeV) states. Schaeffer has also performed similar calculations for Ca40 with proton energies from 17.3 to 55 MeV. He used the Blatt-Jackson potential and Gillet and Sanderson's wave functions. The dependence of exchange effects upon the energy was investigated by examining the ratio of the total cross section 0[D+E] to the direct cross section GED]. A comparison of the results of his calculations with those obtained in this work are given in Table VI-5. F? ....T [- 'i‘xu 131 TABLE VI-5.--Ratio of total cross sections o[D+E]/0[D]. Ep 3 2 22 5' ‘ This? This? (MeV Schaeffer Work Schaeffer Work 17.3 2.7 '6.8 20.3 3.3 7.9 25.0 3.5 7.8 30.0 2.9 3.1 6.4 6.4 35.0 2.8 5.5 40.0 2.5 2.5 4.6 4.8 50.0 2.3 3.6 6.3.3 The 2nd and 3rd 3‘, T=0 States Figure 6.4 shows the results of the calculations for the 6.285 MeV state using direct, exchange and a direct plus exchange force. The experimental cross sections are again wellreproduced except at 40 MeV and at large angles where the exchange contributions are overestimated. The energy dependence of the exchange effects with respect to the direct term can easily be seen as in the case of the lst 3- T=0 state. A comparison of the experimental angular distributions between this 3- and the 1st 3- states reveals some differences xuhich may be attributed to the nuclear wave functions or to the mechanism of the interaction or both. The agreement laetween the antisymmetrized distorted wave (ADW) calculations *Both KK and KB forces give the same results. 0 0’1"»: ...-1. (IDD/ SF ) 132 4°60 (p.p’) LO 2nd 3" T:O —--—KK FORCE E... = 6.558 MeV —— EXCHANGE E‘ 3 6.285 MeV (T.T.S. KUO) —— KK‘EX LO ': :.- \ ‘\ 5,204.: \- .0/ 1 \\—’;‘<./ E \\{\._2 E ,. \.z- - . - - + 1' I i 0.3:}? J— l I I l l J J A 1 1 1 1 1 J l l n 1 n 1 1 20 4O 60 80 I00 20 4O 50 80 IOO ecu (COO) Figure 6.4.--Microscopic DW calculations for the 2nd 3-, T=0 state. ‘6. - . I. ...}..mT.‘ 133 and the experimental results seems to suggest that the RPA descriptions for this state are quite good. However, difficulties were encountered when the ADW calculations for the 3rd RPA 3_ state were compared with the distributions of the 3rd 3- of the experimental spectrum. It was found that the calculated cross sections were 10 times too low, as can be seen in Fig. 6.7. On the other hand, the experi- mental distributions of the 2nd and 3rd 3- look very similar not only in detailed variations but also in the absolute Fina-.4. magnitude. It is possible that the calculations shown in Fig. 6.4 actually correspond to the 3rd experimental 3-. .The extended shell-model calculations of Gerace and Green (Ge 68) show that the 3rd 3- is made up entirely of the 2nd RPA 3- and their 2nd 3- is a mixture of the 3p-3h deformed state as well as the contributions from the lst and the 2nd RPA 3- states. The electric transition rates to the ground state from their 2nd and 3rd 3- states were found about equal (1.9 vis 2.7) when SPE II was used. Gerace and Green's picture is in consistence with the excitation strength measured in this experiment (2.5 vis 1.7). Thus a conclusion can be drawn that the lst and 2nd RPA 3- are good wavefunctions but the 3rd is not. The 2nd and 3rd experimental 3- probably have similar microscopic structures either of which may be described by wg-(RPA). Finally Gerace and Green's theory resolved the difficulties of RPA in giving satisfactory wavefunctions for the 3rd 3- state. 134 6.3.4 The 5‘, T=0,1 States The ADW calculations for the 5_, T=0 (4.48 MeV) state are shown in Fig. 6.5. The exchange term dominates the contribution to give the correct magnitudes of the differ- ential cross sections but overshoots somewhat at large angles. The contributions from the direct term are small as can be seen from Fig. 6.5 and the o[D+E]/o[D] ratio in Table VI-S. The ADW calculations for this state demonstrate the extreme importance of the exchange effect in predicting the correct magnitude of the angular distributions. The results of the lst 5-, T=l state are shown in Fig. 6.7. The distributions of the components LSJ=505 and 515 were found comparable in magnitude. The total distribution is the incoherent sum of these two components. The corresponding experimental results (see Fig. 3.2) Show that the calculations predict the correct normalization. The particle-hole configurations of these RPA 5-, T=0,1 states are mainly [f7/2d3}2], in good agreement with the results of (He3,d) and (d,n) measurements and the theory of Gerace and Green. §;3.5 The Unnatural Parity States The ADW calculations were done for the 2‘, T=0 state (6.025 MeV) at four energies, and for the 2-, T=l state (8.412 MeV) at 25 and 40 MeV. The results are illustrated in Fig. 6.6. At Ep=40 MeV, both T=0 and T=l states are qualitatively reproduced. The calculations systematically 0.I 135 I I I Ij .11] I I ITIIII T 4000 (p. 0’) E, = 4.487 MeV j L 1 1 IO lst 5‘, T= o —-—— K K F0335 = -—— EXCHA E... 4.323M8V Km Ex (T.T.S. KUO) E E,= 35 MeV I L .. l I 111111 I l lllell 1 1 1111111 1 l l L l l 20 4O 60 80 IOO 66" (deg) 20 40 60 80 IOO MSU CYC Figure 6.5.--Microscopic DW calculations for the lst 5-, T=0 state. Ir— 136 .mmumum 2.018 .IN uma msu 20w mCOHuMHsUHMU 30 oemoomouoflzll.m.m whomHm 063 0 2063 0 02. oo. om 8 0.6 02 0.0.0 02. oo. om om ow 02 o 1 .-_O I ,//HHHHHHhk+#5. ..fO L.“ U. n o 0 9+ ++ H m w my 2 . >62 02 m n: l-. _.O W ...l ...-.o- m .2 .. W . 2. 2 2 w 2 2 w -. _.o .- -n_.o . >62 00 L . . .2 s . . 2 2 . . W .m—O w mo_ 2 >22 000.0. mm . . >62 200 .0 ":0 . >622 020 0. 0 >62 2 .620 .60 0 22:2on2n0u0 w 0 22:2_n2.0rs . .....>> ”03. WE. memo... .xwfiv; 0.0 0 (ls/qt“) ("P/DP 137 underestimate the differential cross sections at small angles. The results for the first 4‘, T=0 and T=l states are shown in Fig. 6.7. The predicted differential cross section for the 4-, T=0 state is about 20 times lower than the experimental results of the 5.62 MeV state (see Fig. 4.7). This serious discrepancy has been carefully examined and was found not to be due to any error or mistake introduced in the procedure of calculation. On the other hand, this theoretical distribution resembles in shape to the experimental counterpart. For the 4-, T=l state, the predicted magnitude of the cross section is about 1/2 of the estimated experimental results (the 4-, T=l level at 7.660 MeV was not resolved, but a few clean spectra enabled the estimation of the cross section to be made). It was also noted that both calculated distributions of the 4_ T=0 and T=l are similar. Finally, results for the 6—, T=O and T=l were also obtained as shown in Fig. 6.7. (Note that the labelings are correct.) The experimentally observed 6- or 7— level at 8.845 MeV may be assigned to the theoretical 6-, T=1 state. The assignment of the 9.237 MeV level to the theoretical 6-, T=0 state is also encouraging, because the predicted differential cross sections are close to those of the 9.237 MeV level. 138 .mmumum H.ou9 .Im “H.0na .nv «one . m cum onu MOM mcofiumHsoamo 30 a ma fine .um uma oflmoomouoazln.h.m musmflm j 1 fl 1 I. T w u . "I s 15 so: 83.. . . f n “V H II. I. —00 p l D T I. l I p r OQMEN 1 u v T . I. ) I I l w v n H mm H T 1 _o m I ( I h 6 IITYITW v . >22 58...... . - .smdSQEQ ; . m u h w m . Sec... 6.5. .....3 mex .m..:. J40 30 UEoowOmoE‘ ON. 00. 00 on 0? ON 0 ON. 00. om cm on. ON C q 4 d 4 d 4 q q q 1 q A q q d J u d a d * d u‘ 4 « 4 —o. l lllllLl l ..o .>ozmom.~_ "...... m .m . _ up .-w J r» I“. .oo. >o2 Sim "so n in 6:5. m , I .o. ./o AW .. ’/ ’0 / / / l \ I / I 11A1 llllLLl L 1111 O. i 9 ”a. L111 (Is/aw) vp/op 139 The RPA wave functions of these forementioned unnatural parity states are more or less pure single particle—hole states (see Table VI-l). They are -l RPA lst 2 , T=0 ~f7/2d3/2 RPA lst 2’, T=l ~2p3/2281}2 RPA lst 4‘, T=0,1-f7/2d3}2 RPA 6‘, T=0,l=f7/2d;}2 The similarities in the wavefunctions of the 4- and 6_ states, as well as the differences between the 2’, T=O and 2‘, T=l states are also reflected by the calculated distributions as expected. Gerace and Green's deformed model agrees with the RPA in presenting the wave functions for the lst 4' state. This [f7/2d3}2] configuration has been confirmed by Erskine, Seth gt_§l. and Fuchs gt_3l. in their [He3,d] and [d,n] experiments respectively. Thus the wavefunctions of this state are believed to be well understood. The failure of ADW calculation for this particular state must be due to the effective force used. Perhaps the tensor force will play an important role in regaining the correct normalization. 140 6.4 Discussions on Even—Parity States 6.4.1 §ystematics Fig. 6.8 shows all the even parity states observed in this experiment. The spacing between the vertical lines is in accord with a J(J+l) relationship for the energy of these states so that a graphical inspection for this rela- tionship can be made. The length of the horizontal lines is proportional to the transition strength. The open circles are for those states observed in other experiments (see Table V-l). 40 have been The low-lying even parity states of Ca described in terms of multiparticle-multihole configurations by Gerace and Green (Ge 67; Ge 69) and by Federman and Pittel (Fe 69; Fe 69b). In their earlier paper (Ge 67), Gerace and Green considered some of the low-lying states as mixtures of the double closed ZS-ld shell model state (j=0) with two intrinsic deformed states (containing components with even angular momenta) formed by raising 2 and 4 particles from the 1d shell into the 2p-lf shell. They calculated the 3/2 matrix elements of the Hamada-Johnston nucleon-nucleon force between the unperturbed deformed states and diagonalized the matrix to find the wave functions of the final perturbed states which are mixtures of Op-Oh, 2p-2h, 4p-4h configura- tions. The unperturbed energies of the 2+ levels were -1—m-emqp-r 141 .musmaflummxo Hwnuo sown mmonu mum mmaonflo ammo manu.cw om>ummao r"“Ij'rrTr"'1"V'rfIVVIVVjVIYTr‘r'rf'I ow 2:45 .usmfiflummxm mo ca mmumum muwwmmlnm>m on» no mowumemumhmll.m.m musmwm v 1111. .m ... ~ 9 i.e.? 2. 82.88 825 Emiémfi n (A'W) A983N3 NOIiViIOXB 142 adjusted to fit the perturbed energies of the same to those of the observed levels. The unperturbed energies of J=0 and 2 J=4 levels were determined by the §T(J+1)J relationship with -n2 - 7T ~ 0.1. Their results are: Main Configuration 0... 2+ 4+ 4p-4h (mixed with 2p-2h) 3.55 MeV 3.90 5.25 2p-2h (mixed with 4p-4h) 7.33 MeV 6.90 8.00 where the two 0+ states are further mixed with the (Op-0h)J=0 configurations. The first sequence corresponds to the experimentally observed 3.35(0+), 3.90(2+) and 5.28(4+) states which seem to form a perfect rotational band (see Fig. 6.8). For the second sequence, the 8.00 MeV-level may be either the observed 7.92 or 8.10 MeV level. This 2p-2h sequence does not follow the J(J+l) relationship and no ‘ explanation on this aspect was given. Gerace and Green (Ge 69) also used their deformed model and mixing technique to account for the 5.20 MeV (0+) and 5.24 MeV (2*) states of Ca40. K-band mixing and 6p-6h, 8p-8h deformed rotational bands were included. Their previous calculations (Ge 67) were modified to allow all-out mixing between Op-Oh, 2p-2h, 4p-4h, 6p-6h and 8p-8h configurations. Anderson 52:31. (An 69) compared their (p,p'y) results with Gerace and Green's picture. A k=2, 4p-4h band for 5.25(2+), 6.03(3+) 143 and 6.51(4+) MeV levels, and a k=0, 8p-8h band for 5.21(0+), 5.63(2+) and 6.54(4+) MeV levels were constructed based on the enhancement of the in—band transitions. The former band does not obey the J(J+l) law whereas the latter does (see Fig. 6.8). Anderson et_gl. found that there was a general agreement between the experimental reduced matrix elements and the theoretical values for the 4p-4h and 8p-8h states. But they also pointed out a few discrepancies which demand further discussion. Federman and Pittel (Fe 69), on the other hand, showed that an alternative description for the low-lying 0+ levels of Ca40 is possible which does not require a 6p-6p or 8p-8h state. They proposed a weak coupling model 40 can be in which the energies of the known 0+ levels in Ca accurately reproduced. This model includes only Op-Oh, 2p-2h and 4p-4h configurations, but allows all possible intermediate spins and isospins. The calculated energies for the lst 3 0+ states are 3.29, 5.22, 7.62 MeV, in excellent agreement with the experimental results. The same 40 (Fe 69b). Again model was applied to the 2+ states of Ca all 8 2+ states are well reproduced by the calculated spectrum. It is realized that Gerace and Green's model attempted to retain the band structure of the deformed even-parity states, whereas Federman and Pittel's model emphasized only the configurations of the spectrum of a given even J, thus no calculation was made for 4+ states. 144 These two models have enjoyed success in different areas and a comparison between them can only be made by an experiment on electromagnetic transitions between those states covered by both areas of studies. .So far, all the observed 0+, 2+ states and those 4+ states below 7 MeV have been theoretically investigated. However, the 6+ states and the 4+ states above 7 MeV observed in this experiment may bring new information out of the band structures of the even-parity states in Ca40. Further theoretical and experimental studies on this aspect are desired. 6.4.2 The lst Excited 0+ State There is a general agreement between Gerace and Green (GG)'s and Federman and Pittel (FP)'s models that the 3.35 MeV level in Ca40 is mainly a 4p-4h deformed state. The 4p-4h strength predicted is about 70% by GG model and is about 83% by PF model. The Ca42(p,t)Ca40 experiment (Sm 69) showed that if the ground state is assumed to be a pure Op-Oh (shell-model) state, the 3.35 MeV 0+ state is certainly not a pure 2p-2h state. This evidence complements GG's and FP's results. The configuration of the ground state of Ca40 is described mainly by Op-Oh (~82%) mixed with 2p-2h (~l7%). This mixture has been supported by Ca40(p,d), Ca4o(He3,He4) reactions (G1 65) and also by K39(He3,d) experiment (Se 67). If one compares the wave functions of the ground state and 145 those of the lst excited 0+ state predicted by GG's model, such as Ground state: 0p-0h(0.91); 2p-2h(0.4l) 3.35 MeV state: 4p-4h(-0.83) 6p-6h(-0.45) one finds that the 3.35 MeV state might be predominantly a 4p-4h excitation from the ground state as a whole. In Section 4.7, it was mentioned that the (p,p') data obtained in this experiment could be fitted using an empirical form factor shown in Fig. 4.11, and that the fit is very sensitive to the relative size of the oscilla- tion in the surface. If a form factor—-calculated by using 4p—4h wave functions and appropriate effective interaction-- could be obtained, it would be interesting to see the com- parison between this theoretical form factor with the empirical one. The decay modes of this state have recently been summarized by Harihar §E_gl. (Ha 70). They concluded that the branching ratio of double y emission with respect to internal pair emission is in the order of 4x10-4. The probability for the decay of this 0+ state by conversion electrons is also negligible. CHAPTER VII SUMMARY AND CONCLUSION The angular distributions for protons inelastically 1 scattered from various excited states of Ca40 have been k’ measured at incident proton energies of 25,30,35 and 40 MeV. Data of about 50 states have been analyzed and the systematical " and consistent variations of the distributions with respect to the proton beam energy were observed. The L-transfer quantum numbers for most of the observed states have been obtained and compared with the results of other experiments. Good agreements were obtained in general and some ambiguities that existed in previous experiments were clarified. It is concluded that the (p,p') experiment, performed at higher as well as various proton energies with a good detection system, enables one to determine the L-value with less uncertainty. States with spin-transfer larger than 5 have been observed and identified. The DWBA collective model analysis has been carried out and the deformations 6L's were extracted. It was found that the collective model was successful in predicting angular distributions in agreement with this experiment, 146 147 except for the cases of L=0 and L=l, where it is known to be incorrect description. Generally speaking, the collective DWBA distributions follow the same energy dependence patterns as those of the experimental observations.‘ It also appeared that the overall shape and magnitude of the experimental angular distributions of a given L are roughly independent of beam energy and excitation energy. Therefore, the 6's extracted are more or less energy independent. However, this statement does not apply to every excited state. For example, the individual distributions of the 6.767 MeV state coincide in shape with those of the 3.732 MeV state, but the relative magnitudes in going from one energy to the next do not. Thus the observed energy dependent of 5 for this 6.747 MeV state may be real and interpretations for this pehnomenon are to be desired. The reduced transition probabilities B(EL) scaled for the (p,p') experiment were obtained using Fermi equivalent uniform-density-distributions. Finally, the antisymmetrized distorted wave calcula- tions have been performed for some negative parity states, using the K-K force and T. T. S. Kuo's R.P.A. wave functions. The particle-hole configurations of these states were investigated by examining the overall results of these ADW calculations and by comparing them with other theoretical and experimental results. The nature of the states under study were fairly well understood. It was also found that the central force used in the ADW calculations is adequate 148 in predicting the distributions of the normal parity states, but a tensor force may be essential to reproduce those of the unnatural parity states. Considering the (p,p') reaction in conjunction with other types of experiments as a probe to study the nuclear structure of 40Ca, one finds that the achievement of pre- 40Ca (p,p') experiments was limited. With the completion of this work, the accomplishments of the Ca40 viously reported (p,p') reaction have been much improved and its capabilities enhanced. The probability of further advancement may be high too. A (p,p') experiment performed at beam energy higher than 40 MeV with a resolution less than 10 Kev may be very fruitful. REFERENCES 149 (Ag (Ag (Am (An (Ar (Ba (Ba (Be (B1 (B1 (B1 (B1 68) 69) 67) 69) 68) 62) 65) 68) 61) 66) 63) 66a) REFERENCES D. Agassi and R. Schaeffer, Phys. Letters 26B (1968)703. D. Agassi, V. Gillet, A. Lumbroso, Nucl. Phys. 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All of the plots are shown on the same scale and in the same arrangement so that a direct graphical comparison of the collective theoretical results and experimental data can be made. 158 dc/dn (mb/sr) (JULIE) 0Q r I IIITII IO 0 5 I WWIIIIII 1 IO; 4OCGI (p.p’) DWA COLL. MODEL L=2 111 11111 1 IJLlll da-ldn (mb/sr) (JULIE) 160 ; 40 I l I E : CO (P: p” : : DWA COLL. MODEL- : L=3 1 3 40 MeV . 20 4O 60 80 I00 IZO ecu (deg) usu cvc da/dn (mb/sr) (JULIE) I TTI 4O : Co (9. p’) : DWA COLL. MODEL 3 L= 4 I 1. I 1 ‘I jug—hug L11 1 - i 20 4O 50 80 '00 I20 ecu (deg) MSU cvc da-lda ( mb/sr) (JULIE) IO .LO‘ r r I I r I I I “Co (p,p') DWA COLL. MODEL L = 5 1 1111111 I lllllll L llllllll I IIIIIIII : I- .. 5 25 MeV '0 '5' ‘2 20 4O 60 80 '00 '20 ecu (deg) MSU cvc da'ldn (mb/sr) (JULIE) 163 I I T I I I l I T I I 4000 (p. p’) ’ DWA COLL. MODEL . ...... L: 6 5 .2" \\ IO :- // \ : l/ \\ : /// \ L , \\ % (x 40 MeV ///..— ‘\\\ \"'"-‘h~‘ / \ l6 .- // \\ : ,f \ ' ’4," \ " /..,.-° \ _ \ 35 MeV /”——~\\ / V '05 :- x/ 1“. I /.-/” ‘X we I6 I TUVUTII I \3 30 MeV .,.\ l l l l j T l l llllLl I lllllll j l Illlll I II 111111 I 20 4O 60 80 IOIOJIZIO 9.. Idea) .5. cy. da-ldn (mb/sr) (JULIE) 4.09 . 4o 5 E CO ( p! p’) I: : DWA COLL. MODEL j L = 7 A 5 /’/-\\ IO :- \ -: : l// \ 2 : // \‘ 1 I. \ .. ‘ If \ 40 M V e . $ .//"'—I‘\\ \ '6 F“ .//// \\ \\-/ t .//’ \ 3" - \35 Mev 4‘ \\. ‘ 5 /"—"~ \\ '0 ? //// \\\ \~ _: : ’,,// \\ 30 MeV f . : \ : I \ - , \ I .— -— 05 : 25 MeV E b 7 20 4O 60 80 I00 IZO ecu (deg) MSU cvc dc/dn ( mb/sr) (JULIE) T r I I r I I40 I I I I I I E E CO ( D: p’) Z “ DWA COLL. MODEL : P L = 8 I / \ Z . / \\ q ; / \\ 40 MeV : / \ - 47 \\ ,/2 ...1 \ I6 e’l’ /’ \ \ - C I / \\ \ 1 : // \ T i’ \ . ,// \ .// \ .. ///// \\ '04 __//'/ /’___\\ 35 MeV __ E -’ /// \\\ i b / \\ q % // \ I: / . // 30 MeV - I- // ~ // '04 7” ": E 25 MeV : '04 J l 1 1 J l 1 l 1 L 4 4 l 20 4O 60 80 IOO I20 66M (deg) MSU CYC da-ldn (mb/sr) LG I I I I I I E + I 4°00 (p.9’) P * Ex = 5.899 MeV . - I .0 WI T." I I IIIIII O O O I . I 0.I:- 9 ": ‘r * '=‘ .. 1+0 §§ I. : . , +9 I. .. _ I” 35Mev . - 0.I:- ’ __. A, . . - I I : II’§ +§§I.¢ " _ , I... 3OMeV ‘ O 0.I:- ’ -: 5 *1, : .49”. . - H 25MeV 0.0| I I I I I I I I I I I I I 20 4O 60 80 I00 IZO 90“ (deg) I0.0 )- r I I I I I I I I I q .. 40 I I- ’ «II 9 b . . LO;- ‘- : , : t . - I. I O . - 9 O *- I .. ... 4OMOV‘ .0 ALOr- —. L. : 6 . U) ... ¢ '- \ _ I f ‘T = I- . . c- : ' ’ .’.”’o co ' I 35M6V " \ I b ‘UI.O:" , -: 4; ‘ = I. ,I 9 I - ‘ I” ” ’o 30MeV " :- ... - 0 LG;- , -: . . I .. I." - .. .9 9’9 25MeV I O" I 1 1 L 1 I I I n I 20 4O 60~ 80 I00 I20 da/dn (mb/sr) .0 t TWIIITI T 'I' .0 I a .0 JIIIIT $IITII 0.I E j I I . I I I I I I r I I I 4"Ca (p,p’) Ex 3 8.270 MBV IIIILII I I I 14 * I I H . + ”Ii 40Mev I *III. ‘5 I I . ”I ”I. 35Mev_ I 2 I* I f I 4 - I * ' I H Z'IOMOV1 I I * 1 * 3 ++ + 25Mov: 201410460 so IOOIIZOJ 6.;M (deg) MSU CYC da-ldn (mb/sr) ; r I I I r I I40 I I I 1 E : Ca(PIP') : P . Ex 3 3.900 MeV : O . . I. '3 I- . ’ . I- . . ’ -I LO L— . . . . 40 MOV_: _ I . i ' ° - ° , , . 35 MeV ' LO 3' ' . —: , ‘ ° 30 MeV . LO .- . . '2‘ t , : ° , 25 MeV - I I 0.I 20 40 I20 MSU cvc eonIOo 9‘:M (deg) I.O 0.I da/dn (mb/sr) 0.0 I I IIIIIII I I IIFfiI I IIIIII' I I “Co (p.p’ I I "I Ex= 5.240 MeV I IJIII 0 20 4O 60 80 IOO |20 9,, (deg) dc/dn (mb/sr) I0.0 : 4O ‘ 2 . Ca (p,p') : I- E‘ = 6.905 MeV - . . . . . -1 |.O :— I _: , I 4?. I I I I I -I - ¢ - I I I I.O E- I I ‘ '1 : I 1 - I I ¢ I I . c- . .. I , . ’ LO :- —. e ’ . = J; O . Q I ’ : I- , d . _ b I 30 MeV " I . . . . 9 9 LC :— . -: : I E : . I I O I + : . 9 _ .. ’ , 25 MeV - I 0.I I I I I I I I '1 I I I I 20 4O 60 80 I00 I20 I I I I I I I I 90” (deg) 17me da-ldn (mb/sr) 0.I :- .0 I .0 l O I o b b b b I“ f f Hi I” f M ”+ *+ l J 4°00 (9.90 E,‘ = 7.290 MeV I 40 MeV 35 MeV I I IIIIIII “III 30 MeV I IIIIILII 25 MeV 20 4O 60 80 I00 IZO 90“ (deg) MSU cvc' do/dn (mb/sr) 4o 5 “I. Ca(p.p’) : I Ex=7.865 MeV : ‘ - O.‘ LG.- ‘I '1 I f I Z : +§§ Q : r- .. .’.§ : I 4OMeV I . q I |.O._.- .I '1 I +I I E . 0.. I . . IIII - ' 0 §. 35MOV -< I _ I I LO? ‘ '1 : , 2 : +§QI9. I... _ _ _ I ’I. 30am: .. I .. I 0.I;- *. ': E .....O. 25Mev: L I 0.0I l I I l l I I L I I I I 20 4o 60 80 IOO |20 9cM (deg) da-ldn (mb/sr) l.O - Ex=8.743 MeV P ++I " o.I____ * _. 4; I E .. +ffI : - f . _ I” I I * I o.I____ ? f’I 4OMeV __ 4 I I ' E : ff'I ' I I : - III ff+ . I 0.l:_ ++ I+ 35MeV __ {‘E *III 3 , - , . : f III, ‘ +4} I . 0.|__ * * , 30Mev _ E +I : “MI 3 _ I - _ *I I 25Mev " 0.0I l I l J J I I I I I I I 20 4O 60 80 IOO l20 6w (deg) LG I l I I l I r r I l l I I 4"Ca (p.p’) ' E,‘ = 6.02l MeV 1]”! I I IIII 0.I dc/dn (mb/sr) E- I —: 2 I, 3 ‘I I. . ' . I I . I“ I . If I I 40MeV O.l_— *, I -: E II III '3 I N - § .- I- * I- . III, ’ . i ‘, I 35Mev 0.I?” ., I 1 I ”I 1 A; I I I q ._ . 4 - I,***.‘II . I 30M9V 1 III. 0.I;— __ - I q I ‘ I i ‘ 25MeV 1 ' '7 . .J 0.0' L I I I I I l L l I I I I 20 4O 60 80 IOO |20 ecu(deg) |.O 0.I da/da (mb/sr) 0.I 0.0l I I I I I I I I 4°00 (9. p’) E,‘ = 6.747 MeV .- IIIIIIII I IIIIII I. I I 40 MeV . O . . . O O '0 -o- Hui-l .. I . 35 MeV . I I I , . T: I I i I Z L if, , . . . 30 MeV _ r . O . . O O - O . o . , 25 MeV g 20 4O 60 80 IOO IZO ecu (deg) MSU cvc I da/dn (mb/sr) A .0 lllfiJ‘lTl .. O O .. .. .. l lllllll 177 LO l 1111 E 4°00 (p,p’) I f+”’+ Ex=8.4l2 MeV l l l [17% .. + '0 .. .0 I 111111] I ..f* N, ’, 4OMeV .. l ? 35 MeV I IIIIJIII -. ... .. .. llllllll -O l 30 MeV -o .. -O . l 0.I 25 MeV I IIIIII' .0 l lllllll I l 0.0l . . . 4 20 40 I l l l 60 80 Ibo 9cm (deg ) IZO dc/dn (mb/sr) 178 IO- III IO:- jIII _ b h h b IO I I IIIII I I40 I I I I I I 5 CO (p! pl) . T. Ex = 3.732 MeV : ° . 40 MeV ' , 35 Mev :- . , 30 MeV ° , 25 MeV ‘ I.O da-ldn (mb/sr) LO LO 179 I l l l l I I I 1 I ' I 40 5 Ca( ,p’) : E,=6.28l MeV '_‘ 9°. _ :F—f, o : ’, 4OMeV __ *9... — E o ' i ' '. 35Mev- _ 1H“. '. ... '. 30Mqu _ ff’Io. . _'_ ... 25MeV' I I I 20 40 60 em (deg) 80 IIOIOJIZO MSU CYC I I I I I I I 4°00 (p.p’ E,‘ = 6.577 MeV - '0 - -o- O- O O .4 O 0 q 0 9 q q l I 11411 LO I I IIIIF O O l j - ' 40 MeV l l ah ’- o 35 MeV T I I Ilrl .- 1 1 JLLJJJ V dcr/dn (mb/sr) 1+? ‘ I 30MeV I I IIIII' O O l I 111111 I O l o" . I § 25 MeV I IIIIII' l llllLLl i 0.0I 1 1 1 _1 1 1 1 L n 1 L L 20 4O 60 80 IOO IZO 6‘:M (deg) ; r I I I I I I I I r I I I E : 4oca (pvp’) : u, I + ., q + 0.I E— + ... % ”*N E ; , 35 MeV . IE,-9.l4lMeV ++++ ‘ I t 0.I " I -: m . \ I E {F WM ”u i + .3 - f, - *H 35 MeV - \b + 5,-9.358Mev '0 0.I;- “H ”H H '5 : I a : I + " _ + H 35MeV j L f + H 5,394: IMev_ H (10'? + ‘7‘. ; 35Mev E H E;9.590Mev« 20 4o 60 so. I I60 ‘Izlo 1 90M (deg) MSU CYC l.0 do/da (mb/sr) 0.I 0.0I 40Co (p,p’) Ex= 5.27OMeV lll da-ldn (mb/sr) I I I I4°I I I I I I E C0(p,p’) : E,‘=6.502 MeV - f‘+,’§ . 9 0.I;- 0 , -: : 9 f I + I % {OI +*§ q I I OI— 0+ ,’ , 40Mev_ . § § : I .. 9 i; {fI H 35Mev +II, ’ 0'7 H ,§ ‘5 m, 5 wI H. ' I .9 30MeV_ I H 0-‘5' M *9, "E I 0 * I I. .. *., ZSMqu 9. - 00' 20 40 so so IOO IZO 96M (deg) MSU CYC dV/dn (rub/3r) 184 I I I I ' I ' ' I I ‘ 40C0 (p,p’) E,‘ = 7.454 MeV L llllll O.|:- i+* I -§ : ' 1 . I ‘ 4 ”I ‘ “I 40Mev 0.I:- ‘HH H '3 5 H * - I _ 1 $ ** u 35Mev ._ III — : H+ 3 e - I " " *9. BOMOVd 0.I:— 1****+*+¢ j: I .I q : I 3 . HI, 25Mev- 2‘0L4l01610 BOJIOLOJIZO 9c" (deg) MSU CYC I.O 0.I O O IIIFW dc/dn (mb/sr) 0.I ‘91 I IIIW III I IIIIII l I IIIIIII l 4°60 (p. p’) rsx =7.92l MeV ’ 40 MeV ’ I 35 MeV 9 ’ I 25 MeV ] l l l l l l l l l l Jllllll l l l lJllle n Llllll 20 4O 60 80 IOO I20 9CM (deg) l.0 : 4o 5 : O I 9 9 0 ca (p,p’) . _ ++ '. Ex= 8.36l MeV j I- . .1 9 o" L— . 9 —( % I ’ ’ a - , * 40 MeV I h ’ . _ I) ++ 0 ¢ § . I 4 O _ ’ ’ - AOJ I ' 5 A? ’ ’ I 35 MeV 1 \ I .D b- q E : I’ ‘ I 9 d ) I ‘ a _ + . _ 3 ~ 'UO'I‘E— ’ ' I 30MeV -: ? : -- +§O 9 9 Q , ’ .4 b 0 9 .. 9 o.|_ ’M, 25MeV _ : ’I ; : i - I 0.0; l l l l l J l l l L l l l 20 40 60 80 I00 (20 GW (deg) 4000 (pap’) a 1 0.I 5- )HHI '3 : I 1 + .. f ( i )f 5 0-0' :L H 35MeV ‘5 3 : E,=9.540Me\1 s A; V . '1 .3 . .. \ HH 3 0.I :— ) H —5 I : *)(+ 35MeV " H gage-rem OOI ? @ ‘210L4IOJGJOJBIOJIO‘OJIZBI ecu (deg) MSU cvc I.O 0.I 0.I dc/da (mb/sr) 0.0) 188 L j l l l : ‘0 ’I I j: - Ca(p.p) : “9"“. Ex=5.6I3MeV : {if I -_- l- . .- : **’°"°,. ’. 4OMeV j P‘ . O - II— . — I. I : +9..0.. 35M6V : .f -. 4 ,30Mev 3 )Q.,......o — °°..25Mev ‘ l O 20 40 6'0 60 9c" (deg) I00 I20 '.O I I T I I I I .. . 4O : Ca (p,p’) : ... Q 9 . J 0.I :- ‘ '7- . I - I ° = _ I 3 ' 9 9 O o .7 d . +I’ . 40Mev - 0 °. ’ I a: 0.I E" 9 1 m - I ‘ 3 J; . 3 E " Q, 9 o . - a . ff, ... I 35 MeV 7 IE - o . 9 " 0.I 5- ’ —: Jé " : ... . 30Mev I , _ 9 ' , . - V ' O.l_;- ° . 25 MeV—i E 3 - 1 o. I A l 1 1 l L l l l l I l 0 20 4O 60 80 I00 l20 6CM (deg) da'ldn (mb/sr) 190 I I I I I I I4°I I I I I I E CO (9.9,) : - EI=4.487 MeV : O... J ... C. I. . '0? . '5 . ...-“- . - 4OMeV- +, O . LO:- '. “ lg '. 35Mev - *9........o. LOB- °. if '. 30MeVE L +I0°" "'o. I- I.O:- '. j E ... ZSMCV: 20 4O 60 86 I60 4 I20 ' 0.I .0 I IIIIII .0 I IIIIII .0 I IIIIII' 0.0 I [III da/dn (mb/sr) rfi 1% “F. I IIIIII l {HHH +++H++ l +HHH i§f**+§* "E + 2. l * + 40 MeV Ef 9.025 MeV'f + § + 1 , . ¥ h 35 MeV ‘ # +&9.025Mev_ M *+ h 35 MeV . t *5; 9.856 MeV H 35 MeV : * * + E,=Io.045 MeV - llllll l J l l l l l 20 40 I00 I20 MSU cvc 60 80 9cM (deg) 192 dc/da (mb/sr) 9 9 9 IIII I I I I 1 I I I I l T l 4oca (p,p’) L l llllll n LL 1411' l lllll I W” + H H‘ 35Mev - '1 HH 2 H * +H H 25MeV : E,:8.I88Mev: q 4 l L 20 410160 so Iolollzol 9m “69’ MSU cvc .X' 1'.- r I I j F I 40on (p,p’) - H ' ’ * . 9 . E,.=8.847Mev 0.I: §I .. 5 f‘ . 3 I ’ ‘ % , 40Mev: I I 0.I.— H “’ H. _ .1 : *I I Z: I ‘ l . O 35MCV. ’2 it .. In \ q .n a '5 *9 ’9. soMevg g : 1 : b . . '0 J? * I §**+ 0.I;- H 9* *H. zsmv-g 0.I . E +**’*”*+ 5 .. ** q ’ I 35MeV‘ - IE;9.237MeV 0.0| l l 1 1 1 1 1 1 1 1 1 4O 60 80 IOO IZO ecu (deg) MSU cvc 1 77" da'lda (mb/sr) 0.I 0.I I IIIII w I I T I —r ' r I I I I 4°Ca (p. p’) E,‘ = 7.670 MeV lllllll * + + 4- l . 40 MeV O I lllllll J 35 MeV 'o. 3OMeV —+._ ... .. ... .. .. .0 . C Q C . C C .. O C C 1 . 11ml “L J .'o.'0'o. ZSMCV 1 1111111 l so so IOO 1:20 I ecu (deg) MSU cvc 20 4O APPENDIX II TABULATED ANGULAR DISTRIBUTIONS 195 QQCTUK 9cmn (vs/9w) 4457-ECCC 1571-2033 5C9-37q5 163.2333 Ike-E372 5096“; 20-1313 72.664; ICh-39AQ 900531; 6&03943 310715; 1505259 11.38%; 19093?3 1'3-77‘9 19-2743 12- 389:; 1006853 £07471 73fT”K ELASTIC 36700 (Va/5R) “SEE-2996 1591-5053 BER-769$ 136-1903 123-3893 190343Q 53-4513 351-3003 lCe'ISCD 7?}- 391:3 39-946Q 1?.9RO? lCoflgQJ 9-65?3 907592 ll-lOoq 1:069?3 ‘04576 b-b7IH 391496 Ense- (X) 00353 C0105 0'120 C'3PB 3.242 00805 3-434 00254 0-202 00204 0.228 00250 00349 0'415 Co##4 00423 00378 0-320 C0322 0'339 SCATTERINQ A:G(L-) (DEG) 12-00 17-00 '22-00 P6-7O 37-03 31-70 36-70 41-70 47-00 52-00 37-00 $1.70 66-70 71-70 72-00 77000 82-00 86-70 91-70 96-70 A‘\G(L.) (DEG) 12-00 17.00 92-00 26-70 27-00 31-70 36-70 41-70 #7-03 52-00 57-00 61-70 66-70 71-70 72-00 77-00 82-00 36-70 91-70 96-73 ELASTIC SCATTERInG AT 24-926 MEv DGVDIL (MB/SR) 4685-8008 175#-974# 638-9622 170-840# 153-4668 16-6738 27-2669 81-7283 110-1534 98-5710 66-1162 38-6321 18-9903 11-2687 1100998 1009068 12-35#6 12-1133 10-6655 8-6868 AT 300044 MEV 06709 (MB/SR) 4241-0781 1776-6917 617-1724 1#2-5841 126-0068 20'1968 57-7913 105-7859 109-9363 74-7065 #0-0558 20-“864 10-3035 9-7968 9-9174 11-2fi46 10-7620 8-#696 5-57#O 301369 197 ctec DRerww ELASTIC SCATTERING AT 34.775 MEV A:'.'_’3(Ci"') DG/DQ ERRSR A:uG(L-) DG/Dfl (DEF) (MB/SQ) (z) (DEG) (MB/SR) 11.?r 4073.399? 3.112 11-50 #284-5156 15.03 1904.099? 0.092 16-50 2000-6177 RB-CS 669.36“? 0.136 121-50 702-#241 96.?6 155.8873 0.329 P6-20 163.3601 31.02 23-9800 0-431 31-20 30-2831 370C; 67-757J 0.268 36.20 70.6285 “201% 104.3933 0.173 41-20 109-Ofi34 47-28 94-0353 0.153 46-20 97.4852 52.?7 59.3363 0.174 51-20 61-2902 57.64 27.7533 0.251 56-20 28-5458 6?.81 12.2363 o.357 61-20 12-5235 67-57 3.6144 0.389 66-20 8-7833 7?.6? 8.9522 0-296 71-20 9-0970 70.9? 9.0077 0.506 71-50 9.1453 77.92 3.9908 0-520 76-50 9-0970 92.99 7.1135 0-530 81-50 7-1666 R7.60 4.8260 0-387 86-20 #-8501 92.7: 2.9197 0-515 91-20 2-9197 97.69 1.9197 0.617 96-20 1-6650 CAAC DHPTBV ELASTIC SCATTERING AT 39.§28 MEV ARCH“) (ac/r30 ERRB'R A.~.G(Lo) 0070!) (0E0) (”B/84) (x) (DEG) (MB/SR) 12-31 3959.5999 0.069 12-00 “060-5928 17.14 1662-8997 0.090 17-00 1767-4944 22-56 526-0296 0-116 22-00 551-9268 27.38 115.7503 0-286 26-70 121-2291 27.68 95.3863 0.218 27-00 99-8982 32.40 44-1820 CokOQ 31-70 46-1848 37-60 84-7930 0.216 36-70 88-3882 42-79 93.8933 C-165 41070 9706058 69.10 63.7763 0.255 47.00 56.0579 5?.19 30.3310 0-287 52-00 31-3207 59.26 12.668: 3.412 57-00 13-0302 63.52 “.0363 0-543 61-70 8-2042 16,3. 7.6443 5.430 56-70 7.7940 73-]? 7-2943 0-409 71-70 704079 73-A3 7.2460 0-548 72-00 7.3596 79-47 5-6030 0-534 77000 506705 93-6? 3-1540 0.576 92-00 3.1852 99-27 P-GOQO 0-785 86-70 2-0028 9?.2: 1.323; 0.828. 91-70 1-3271 99.7: 1.062? 0-746 96-70 1-0617 198 7AACCA40- Ex=.3-3SO WEV E0: 34.996 rEv EP= 30-941+ ”EV AI.G(C"’) 207012 Ewan: A\G(CM> 06/017. ERROR (or?) (“?/§R) (z) (050) (MB/SR) (X) 27.4? 3.1353 13-7c 27-41 0°?000 15'00 32.54 3.3205 13-59 32.53 0.0760 15.53 37.45 9-3743 13-14 37-65 0.0635 11-18 42.76 9:72q 1135; 42.76 0.0625 10-08 48.16 3.3785 13-23 48.16 0-0692 10-40 53.76 ..3713 12-82 53.24 0.0520 10-38 58.34 6.0600 12-17 58.32 0.0336 12-50 63.39 c.3502 9-36 63.38 0-0229 11-79 68.?1 ;.0410 9.27 68.14 0.0166 12.05 73-21 C-LBQE 9033 73-19 006176 13007 78.55 0.3253 13-83 78.5» 0.0200 12.00 83.5“ 3.3193 14951 83.56 0.0176 13.6- 28-29 ;.~1AC 13357 88.27 0.0140 11.43 930?") Q- 14'“? 98027 009065 15038 .p- 1.775 “LV EP: 39.623 MEV ms ( C“) :0700 5137391? AMHCM) 007012 ERRBR (hrs) (' (Z) (DEG) (MB/SR) (X) 26.9: 23°81 27-#1 0-0678 16-52 32.,p 11-98 32.53 0.0396 16-41 37.1? 17.42 37.54 0.0372 15.86 42.21. 1m5~~ 42.75 0.0639 11-16 1.7.123 «2.14 48.15 0.0415 18-07 52.9,? 1M2? 53.24 0.0369 12.20 =57.51 11-72 58.32 0.0227 14.10 62.50% 12-22 63.08 0.0140 15.00 67.6“. 13-25 68.14 0.0075 18.67 72.6%: 11-57 73.18 0.0080 20.00 75.5fl3 20-15 78.53 0.0075 21-33 57.77 12812 83.55 0.0060 20.00 92.77 14.71 88.27 0.001.. 22.73 577.276 15-22 93.27 0.0035 20.00 - 98.96 009026 23008 RP: 24-926 MEV AMHC") (DEE) 12-33 17047 22-60 97.4? 27-7? 32-54 37-64 “2-77 48-1“ 53°?7 58.3% 63-11 68-19 73-25 73-5? 78-57 53.59 88-3? 93-31 98-30 ‘— CF= 34.775 ”EV 199 CA40(D)P')CA40* EAHav (%) 3-00 1-78 1-01 1340 1303 1-10 0-79 0°82 3°83 0°85 0982 0373 0‘6? 0‘6? 0-70 0-70 0-71 0'66 0-70 3-76 E9H59 (%) 337? 1344 1-06 1-18 3'63 3-66 0359 0°59 0f63 0371 Ex: 3-732 EP- 30.044 MEV ANG(cM) (DEG) 12.33 17.47 23-60 27-42 27-72 32-b4 37-65 42-76 48-16 53-25 58-33 63-10 68-16 73-21 73-51 78-55 83-57 88-29 93-29 98-28 EP A\G(cM) (DEG) 12-33 17-46 22-59 27-41 27-72 32-53 37-54 42-75 98-15 53-24 58-32 63-09 68-15 73-20 73-50 78-5“ 83-56 88-27 93-28 MEV 00700. (MB/SR) 8-3030 10-9910 11-8650 11-7610 11-7390 11-4060 10-0450 8-5970 6-4840 4-3890 4-0630 3-9330 3.6800 3-9390 3-“340 2-9940 2-3550 1-6820 1-1560 O-§76O 39-328 MEV DOIQQ (MB/SR) 9-é300 12-5700 14-2300 14-3720 14-3990 1304110 10-9000 7-6900 4-8380 3-2600 2-8860 2-7940 2-3090 1-7900 1-8030 1-2260 0-7750 0.5940 0-9970 ERROR (X) 2-60 1-70 0-86 1-16 0-62 1-06 0-65 0056 0-64 0-67 0-78 0-62 0-59 0-64 0-69 0-67 0-83 0-75 0-83 0-99 ERRBR (Z) 1-87 1-24 0-75 0-78 0-55 0-73 0-60 0-57 0-92 0-86 0-87 0-73 0-78 0-83 1-10 1-14 1-16 1-44 1-36 200. CA4C c.3090 13.33 88.31 0.0078 16.00 92.5y; 0.5084 1430: 93.31 0.006“ 15.50 CA40* Ex= 5.270 MEV PP: E49996 Nev EP. 30.944 MEV A:~.‘G(C"') 06/0!) Ewe-Q Ar-szM) 00700. ERRBR (DPS) (NB/SR) (Z) (PEG) (MB/SR) (X) 17.50 0.1800 2003: 17.08 0.2100 20.00 22.63 3.1050 15903 22.01 0.0960 15.00 27.45 0.105: 17:50 27.74 0.0980 9.80 32338 CHIC-0"} 14fl+Q 32'56 000970 14080 37.70 0-092? 11~00 37-68 0.0920 11.00 02.91 3.0980 1102: 48-?0 0o0830 12.60 «8.22 0.0820 103.: 53.29 0.0700 12.70 53.31 2-0730 12340 58.37 0.0430 13.80 58.00 c.c630 12900 63.14 0.0330 11040 63-17 0-0540 IOfSO 68.20 0.9320 ‘9.30 58.2? 3.5450 9-50 73.56 0.0270 11.00 73.58 3.3360 10.83 78.59 0.02u0 12.90 78.62 0.3320 13680 83.52 0.0200 12.00 83.65 2.0300 12-20 88-33 0-0155 16-00 88.36 0.2260 12520 93.33 0.0135 17.00 93.35 3.2230 10.30 98.33 0.0125 14.60 98.35 0.02C" 10°30 :9: 34.775 va EP- 39.§28 MEV A0 ( c") fro/m mm»? A~.G DO/Dfl ERRBR (0:5) ( E/SF) (%) (DEG) (MB/SR) (x) 11.5? 3.1200 30cc: 12.33 0.1200 25.00 16.95 F-185O 25-00 17.45 0.1800 20.90 22.10 2.3800 P2300 22.00 0.0750 16.20 25.9? 0-0780 15°03 27.73 0.0780 12.50 32.94 C°C79O 13.09 32.55 000780 14060 37.10 2-0780 14é03 37.65 0°°73° 1"00 42-27 0-0740 13640 42.77 0.0660 15-50 «7.37 :.\A30 12.40 48.18 0.0540 16.80 52.46 - r540 12300 53.27 0-0400 16-80 57.5% “.3410 11350 58.35 0.0270 14.00 6206‘? L,r\275 3200C 63012 009190 17060 62706? $05321“ IE'OC 68018 000140 14090 73.94 FwCl7C 14-53 73.53 0.0110 17.20 78-02 0-316C 15-0: 78.57 0.0095 15.00 g;3.1c g.c140 15f03 83.59 0.9087 14.60 a7.qs: 0.3120 12300 88.31 0.0070 16.00 92.92? 0-0109 14300 93.31 0.0060 15.50 97.31 3.3064 15340 98.31 0.0047 17.60 ED: Akfi‘CM) (DEG) 12035 17050 3206? 27046 27-76 32°59 37071 4208? 48-2? 530?? 58041 63019 68025 73-30 73060 78064 83.67 8E03? 93033 98.37 59: .AfiG(CV) (DEG) 11'97 16096 22010 26.9? 32":'5 37016 42027 47038 52047 57-56 620‘3 67069 72074 730C5 780:9 33°11 0703? 92093 97-89 CL4O(DIP')CA4Q* 240996 ‘P 00 / an (01/35) @0237? 3.298” C" 3553 003514 3'3'53‘2 0°335T 5'422? 0-45#¢ $04g72 C04Q95 c-aa15 0-3910 3.3573 Co?892 Q-3026 309451 0?123 -1%c9 "161? -1 c\ R "\ a -‘ C)D(§(3 34.775 W) 204 ERRPQ (7) EQRQQ (%) A0°00 19'00 10~00 7§94 4341 4307 3f33 2°83 268: 2375 2'84 3?07 2°73 4083 5693 6322 3'70 4°25 “‘61 Ex3 50613 HEV EP- 30.044 MEV ANG(CM) (DES) 12034 17048 22062 27044 32057 37.59 42050 48021 53.30 58038 63015 680?1 73027 73057 78061 83063 88035 93035 98034 EP: 39.§28 MEV ANG(cM) (DEG) 12034 17047 22050 27043 27074 32055 37067 42078 48018 53028 58036 63013 68019 73024 73054 78.58 83-60 88-32 93032 98031 UJIMQ (MB/SR) 003200 003800 004300 004836 004291 004193 004005 0.3566 003226 002880 002430 002090 000610 00g710 001370 000996 000735 000610 000522 00700 (MB/SR) 002750 0.3280 003500 003790 0.3755 003567 003353 003406 0.3170 002529 0.2053 0.1550 001054 000704 000731 000498 0.0402 000330 0.0268 0.0198 ERROR (X) 21050 14070 6064 6013 5050 3038 2092 3031 3006 3034 2065 2051 3015 3032 3037 4043 4006 4036 4064 ERRGR (X) 22085 14035 10000 5060 4024 4093 3090 3030 4011 3058 3051 4030 3098 4058 6013 6055 5075 6060 6028 5090 3| r9 AHG(CN) (0??) 12.35 17-5C 22064 P7046 07.77 3205” 37071 “2083 48.24 53034 58.4? 63010 68026 73-31 73-61 78.6% 33069 88039 930?Q 98.3w 5?: CA4C(DIP')CAQO* 340996 “EV :10/ DD. (”B/SR) 0°4500 (70430". 004150 7°3043 3.3320 002113 301569 3°1057 303325 0-3713 COCSSC 303505 J'CQ31 303490 3.3461 303476 003457 £00420 0:318 033C? U(J 340775 VEV 00700. (”5/55) 3.5903 0'5400 004100 002997 001425 C09969 30:682 0'0515 000389 306301 ~3321 000430 ”05339R U ( U o ) U m U H()()U O O N’ V w 205 ERHa? (%) J ULW‘QO‘UsU.nOJ ,..0 L41“ m|NL)U U1wCP$WV¢u\JO :(uu .r. ... . . ... 8V ERRSR (K) 23'89 11°03 1035C 9002 7330 8375 9'52 10.30 8-84 13°33 3357 6954 Sf35 12345 13'35 13388 5°70 6’67 5°95 Ex: 5°900 EPI 300044 MEV ‘Dcvmn (MB/SR) AmG(cM) <053> 12035 17.49 2206? 27-45 27075 32057 37069 42080 48021 53031 58039 63016 58.22 73028 73058 78062 83064 88036 93036 98035 SP: 39.828 MEV AxG(CM) (DEG) 12034 17048 22061 27003 27074 32055 37067 42078 48019 53028 58036 63013 68019 73024 73.55 78058 83.61 88032 93032 98031 MEV 005200 000600 004100 002790 002634 001684 001112 000724 000532 000426 000370 000341 000375 000405 000407 000373 000257 000227 000220 000224 00100 (MB/SR) 006500 0-5600 002650 0-1900 001860 001203 000810 000573 000460 000416 0.0370 000276 000344 000294 000277 000186 000215 000229 000220 000159 ERRBR (X) 38000 15080 7020 7094 5010 8077 7030 8060 13010 12023 7065 9072 7002 7092 7026 7014 9067 8016 8000 7075 ERRBR (X) 17060 12090 9028 8011 6064 9018 7011 9092 9066 11093 7028 12055 7044 7048 11092 11062 8027 8025 7000 6059 206 CA4OCD:P')CA40* EX= 60021 MEV EP= ?4.926 “EV EP' 30'9““ ”EV MGM”) DU/DD. ERRsR ANG(CM) DO/DQ ERRBR (DES) (ME/SR) (7.) (DEG) (MB/SR) (X) 12.35 “.180' 64309 12.35 0.1590 51.40 17.5. 0-1650 3C°6C 17.45 0-1800 34-80 22.64 2.1350 15960 22-52 0.2060 10-50 97.4.: c.2004 13303 27.76 0.2160 5-80 27.77 ;-195° 9374 32.57 0.2103 7-67 32.5:Q b.1914 37-97 37.59 0.1940 5.06 37.7? 3.1780 5&92 42.21 0.1670 4.61 42.83 1.1750 6é30 48.22 0.1280 5.75 48'?“ 301667 6'88 53031 001020 5083 5303A T'1F15 5377 58040 000882 6019 55.43 3.142" 93088 63.16 000809 5036 63.20 0.1372 Bios 68.23 0.0783 4.52 68.28 3.1320 3°88 73.28 0.0734 5.15 73-31 2.1305 4325 73.98 0.0741 5.18 73.5? J.1300 4347 78.62 0.0665 5.13 78.66 3.1160 4?51 83.65 0.0520 6-27 83.68 3.1576 “ass 88.36 0.0413 5.62 35.40 3.0223 4324 93.36 0.0271 8.00 93.40 o.c691 4651 98035 000168 9000 95.3a 3.3.5? 5324 ’ EV: 34.775 “EV EP= 390828 "EV Axe ( CM) t-U/Dfl EQRQR MIN 3.“) DO/DD. ERRBR (0:5) (M3/3R) (z) (0E3) (MB/SR) (x) 11.8? (3.1753 73.01:, 12.34 001620 455050 16-97 0-1990 28583 17.48 0.2020 24.00 22.10 6.2150 17.0: 22-61 0.2130 12.45 26.93 ;.2357 10-40 27.43 0.2230 7.50 32.05 3.2287 5356 27-74 0.2320 5078 37.17 0.2007 5°61 32.56 0.1815 7.18 42.29 2.1505 5972 37.57 0.;607 5.62 ¢r7.39 :-1180 5-25 42.78 0-1278 5.86 52.48' 3.3900 5345 48.19 0.0842 8.06 537.57 o.c677 6§o7 53.28 0.0592 7.78 ¢2.eu+ c.3643 5-42 58.37 0.0476 8.18 67.7T‘ c.c547 5675 63.13 0.0421 9.20 ‘72.75> 3.0475 4;75 68020 000362 7020 73.3% 3.3461 8663 73.25 0.9289 7.70 76.1“ (720339“ 9:63 73.55 009291 11000 .83.17> 3.3?90 10-16 78.59 0.9215 10.86 .27.:24 ;.3243 5-13 83.51 0.0109 12.95 92.21. 3.3164 5035 88.33 0.0075 16000 SW7.%€? 0.3104 9375 93.33 0.0068 12.86 98.32 0.0056 11.07 207 CA40(D.P')CA40* EX= 60281 MEV 5p: 24-926 “Ev EP- 30.044 MEV MGM”) 30700 E13853. ANG(CM) DU/D-Q ERROR (093) (03/99) (6) (PEG) (MB/SR) (X) 12035 101800 11°C: 12036 100800 8000 17.82 1.2100 5.58 17.50 1.1500 11.00 22.64 1-2300 3.53 22.64 1.1700 6-00 37.47 1.1473 4-16 27.45 1.1506 4091 32.60 1.1170 3- 2 32.58 1.1285 3.39 37.7? 1.9057 2-31 37.70 0.9686 2.18 42.84 3.9195 9°43 42.81 0.8125 1.94 48.35 3.8175 2.57 48.22 0.5508 2.37 53.3.: 307043 2965 53032 004519 2055 58.44 0.8301 2778 58-40 0-3008 3'10 63.21 0.4248 3.53 63.17 0.2717 2.63 68.27 3.3431 2370 68.34 0.2537 2.35 73-33 3.2703 2078 73029 002356 2059 73-63 302780 2'94 73059 002361 2090 78.67 0.2171 3816 78.63 0.2254 2-62 83.70 3.2154 3034 83.56 0.1949 2.99 28-41 3.161? 6.99 88.37 0.1446 2.77 93.41 3.1395 aéoa 98.36 0.1084 3.0. 98.40 0.1310 2395 PP 34.775 va LP: 39.828 MEV ,1:st ( CM) $0709 EQRQR ANG ( CM) 00700. ERRBR (023) (83/38) <%) (053) (MB/SR) (X) 11.33 0-9600 12933 12.34 Oo§052 9041 16.97 1-0800 6900 17.48 0.9335 5.54 22-11 1.1200 4°25 23.61 1.1081 3.34 26093 101224 400:) 27043 101487 2096 32-36 1~1020 2§40 27.74 1.1678 2-18 37.17 3-9540 2-39 32.56 1.0495 2.75 42.29 3.7104 2328 37.58 0.§085 2.33 1"‘7'39 309965 1°81 (+2079 0053‘?“ 2050 52.4o 0.3075 2376 48-20 0-3360 “'03 57.57' 9.2521 2493 53.29 0.2271 3.77 53.5£5 3.3559 3.55 58.37 0.2841 2094 67071 (30286“- 2'26 63014 002923 2098 72.76- 2.2600 1382 68.20 0.2760 2.35 73.96. c.2671 3511 73.25 0.2385 2.36 78.1C) 0.2419 3-39 73.55 0.2443 3.14 83.1fi3 0.1948 3.42 78.59 0.2009 2.95 87.8?» 3-1514 2025 83.62 0.1568 2.71 92.??3 0.1316 2953 88.33 0.1288 3.17 97.3y. 0.1074 2-54 93.33 0.1032 3.04 98.32 0.0768 2.86 208 :440(°.P')CA40* Ex= 60502 MEV 52: 24.926 “Ev EP: 30.044 MEv 0.81:0 90700 1.2888 MG 5°00 436? 3300 4°90 4'83 4'80 507C 6330 633C 5°40 4'80 #580 4660 5°13 5°23 “'89 0670 4053 Ex= 5°94# EP: 300044 MEV ANG(CM) (DEG) 12035 17049 22063 27046 32059 37071 42083 48024 53034 53042 63019 68026 73061 78065 83058 88039 93039 98038 SP: 39.028 MEV Aéui(cM) (DEG) 12034 17048 22062 27044 32057 37069 42080 48021 53030 58038 63015 68022 73027 78061 83063 83035 93035 98034 MEV 007 00. (MB/SR) 200000 301300 207900 109600 103000 008700 0.5900 004700 003700 003200 002950 003050 003200 003100 002700 002200 002250 002150 DOV on (MB/SR) 203600 206900 109400 103400 003140 006100 004000 003350 002700 002250 002280 002560 002380 002140 001910 001720 001500 001300 ERRBR (X) 5010 3080 2070 3060 3060 3020 3010 4050 3080 3060 4000 3040 3050 4030 3060 4000 3050 3020 ERRGR (X) 6080 4080 3040 5040 5040 4080 4070 6070 5070 5040 6000 5040 5040 6030 5040 6000 5020 4080 LP: 24.9?5 VEV MLHC“) (DEG) 1203‘ 17051 22065 27020 3206? 37075 42037 45020 53.3? 58047 63024 68-31 73067 73071 33.7? 28040 93040 98044 RP A\G(C”) (DFJ) 11m 16097 22'01 26094 32.07 3701q 42'30 47041 52051 57059 62067 67073 72071! 73009 178013 53014: 870547 5%3'87 577056 CA40(PIP')CA40* Doyan. (Vfl/SR) 2.1908 £02160 C'2500 003060 303545 3-3935 304000 304137 [04158 0.0392»R 00337? 302900 002285 301958 001416 301138 3.3974 f0 000550 340775 30709. (\fi/SR) 213 ERHQR (X) 54903 27~4C 12390 8346 5-33 3380 0032 3387 3'57 3f35 3°00 4366 3324 3§83 3996 3-73 3°75 3°85 EHRGQ (%) 60f00 32300 15700 13°76 4°88 4748 4°05 3°28 3f33 3°41 3349 3-5? 33?“ 5990 $383 6°87 4385 6f35 bf73 Ex= 7.110 EPs 30.944 MEV AVG(CM) (DEG) 12035 17050 22064 27077 32059 37072 42083 48025 53034 58043 63020 68026 73032 73062 78066 83069 88040 93040 98039 EPS 390528 “EV ANG(C“) (956) 12034 17048 22062 27044 27075 32057 37069 42080 4802 53030 58039 63016 68022 73027 73057 78061 83064 88035 93035 98034 MEV DGVDfl (MB/SR) 002150 002450 002900 003240 003450 003214 003223 003040 002570 002420 0.1910 001620 001310 001307 000988 000745 000549 000425 000345 DUVDCI (MB/SR) 001630 001800 002040 002412 002485 003590 002560 002570 002370 001638 001436 001114 000810 000642 000618 000448 000354 000287 000146 000132 ERRBR (X) 42000 24000 8085 4040 5084 4016 3033 3050 3040 3058 3013 3012 3070 3094 4024 4070 4087 5040 5080 ERRBR (X) 44000 22052 10092 7010 5060 5090 4032 3077 4055 4049 4032 5004 4060 4070 6075 6083 6012 7008 9067 7026 214 CA4C(P0P')CA4O* EX' 70290 MEV FP: P4.9?6 “Ev 5P3 30°04“ ”EV AF\G(CM) {TU/0n ERRaR ANG(CM) DG/Dfl. ERRBR (EEG) 4vQ/SP) (z) (053) (MB/SR) (X) 27.80 0.3290 14°50 27077 000720 21000 32062 3.3 00 23°00 32.59 0.0500 24000 42.27 3.3530 15-00 37.72 0.0350 22.00 480?“ 9.566? 14030 42083 000390 17'00 53039 UoQEEC 16°00 48.25 009350 21°00 63024 0.3?70 15°50 73.32 0.0210 13.00 78.71 3.5200 10°00 78066 0.0150 16000 £3.73 3.5190 17°03 83.59 0.0120 22000 85.4% 0°0180 11°03 88.40 0.0081 20.00 98.44 300150 1235: 98.39 0.0052 21.00 RP 340775 EP= 390828 MEV chCV) TAO/DD. ERRED? ”5‘0”" DGIDQ ERRBR (Ora) (MR/3R) (x) (DEG) (MB/SR) (X) 26.94 0.3950 23°00 27004 009790 17000 37.16 Q.C4C3 15°30 32.57 000460 20000 02.3: 0.3270 ?2003 37069 0.0350 15.00 47.41 C.g?10 21°03 42.50 000260 18000 02051 {-3180 23°00 58039 000150 18000 57059 C0317D 21.03 63016 .000130 18000 67.73 :.c150 13°00 68022 0.0100 16.00 720753. 003130 14003 73027 000084 16000 78.13 3.3120 23000 78.51 0.0060 16.00 23016 0000212 25.03 83064 000053 20°00 57.27 7.3071 14.03 88.35 0.0056 17000 92.37 :-2065 16°00 93.35 000036 22.00 97.2A 3.005? 16°00 98.3“ 0.0027 20.00 215 CA40(D.P')CA40* EX= 7.454 “EV {2: 24.926 MEV EPa 30.044 MEV «10(0V) “0x0 LRReR ANG(CM) 00700. ERRBR (CF') “/SP) (%) (DEG) (MB/SR) (X) 17.52 3.3708 no.0; 17.50 000700 50000 33.54 3.9741 25.4; 22.6“ 0.0764 16070 27.80 2.686: 26~30 27.78 0.0840 16.20 32.62 3-9955 15320 32.59 0.0953 12.60 37.7% 0-1073 10:00 37.72 0.1080 8.50 42-87 2.1125 8°50 «2.82 0.1090 5.90 48-3T 5-1130 8-93 48.26 0.1060 7-10 53.40 0.108: 7o50 53.35 0.0809 7.30 58.a= 0.1051 7343 58.4# 0.0684 8.10 63.2! 5.3864 6'40 63.20 0.0565 6.20 68.33 2-0717 6§80 68.27 0.0530 5.50 73.68 0.3660 7o30 73-63 0.0848 12020 78.79 2.3552 7330 78.67 0.0360 7.40 83.75 0.0160 a-ac 83.70 0.0307 10.20 88.47 2.0415 6990 88..1 0.0280 7.20 93.a7 3.0926 6°13 93.42 0.0246 7050 98.0% 3.3381 0353 98041 009213 7o#0 PP: 30.775 “Ev EP: 39.828 MEV 1.") 30700 2:28;»: A*.G(C“‘.) 00/01'). ERRBR (0:3) (ha/$2) (z) (DES) (MB/SR) 1%) 16.98 9.3760 31.40 17.49 0.0810 27.00 22.11 0.3839 95-00 22.62 0.0890 18.30 26.9u 3-0943 22690 27.76 0.1010 8.10 32.07 0.1090 10360 32.58 0.1110 10-10 37.2. 9.1180 8-53 37.71 0.1170 8.00 «2.31 0.1110 6690 12.32 0.1060 7.30 47.41 0.0994 6670 48.22 0.0930 8.50 52.51 9-9727 7320 53.31 0.0612 8.50 570:0 0031173 3'13 58040 000391 13030 62.67 0.0410 6460 63.17 0.0290 9.40 67.74 0.0365 6360 68.24 0.9265 9.50 72.80 3.3335 5390 73.58 0.0210 12.30 77.83 0.0305 11.20 78.62 0.0184 11.30 82.66 0.0270 9o30 83.65 0.0170 8050 87.88 0.0248 6o.” 88.38 0.0157 10.00 92.2» 0.0175 7§79 93.36 0.0126 11.00 97.82 0.0132 3310 98.35 0.9102 9.30 -216 CA40(P:P')CA4G* Exs 7-539 NEV RP: 94,926 MEV EPI 300944 MEV 411.0(0) 913/00. FARM ANGmM) 00/09 ERRBR (are) (40739) 1%) (DEG) (MB/5R) (1’ 12.37 0.3000 32.34 12.36 0.2800 28.67 17.52 :-3350 19300 17.50 0.3140 19.30 22.64 0.3423 9-74 22.64 0.3350 9.05 27.20- 0-3800 6639 27.78 0.3420 4-66 32.62 0.3709 7360 32.59 0.3440 7.11 37.75 3.3500 3-97 37.72 0.3260 4.12 42.86 0.3340 3-96 42.82 0.2980 3-50 48.30 0.308? 4312 48.26 0.2620 3.98 53.40 0.2850 3589 53.35 0.2010 4023 58,“. 0-274? “:05 58.44 0.1580 4-68 63.26 0.2482 3355 63.20 0.1330 3.94 68.33 0.2163 3962 68.27 0.1030 4.04 73.68 5.1200 3371 73.63 0.0810 5.61 78.72 0.1540 3396 78.57 0.0680 5.10 83.75 0-1320 3343 83.70 0.0550 6017 88.47 0.112: 3975 88.41 0.0451 7.07 93.47 ,.1007 3.74 93.42 0.0407 6.80 98.46 3.3865 3f73 98.41 O-g309 6'28 EP 34.775 “Ev EP. 39'§38 MEV 41.5101) $0730 2.3593 M;G(CM) 00/00 ERROR (egg) (fig/3R) (X) (pEG) (MB/SR) (X) 11.9? 0.2903 5701# 12.35 002750 32025 16.98 0.3370 19650 17.49 0.3100 19-58 27.11 0.3513 12996 22-62 0.3370 8.23 26.94 0-3590 7-94 27.76 0.3500 4.72 32.07 9-3609 4638 32.58 0.3620 4.76 37020 5031613 5°02 37071 003050 #013 42.31 0.2510 2992 42.22 0.2280 4.00 4+7le 00198?) 4302 48022 001860 5060 52-51 c.1530 4°40 53.31 0-1400 4-90 57.50 0.1320 4318 58.40 0.0960 5.52 62.67 0.1050 4-09 63.17 0.0723 4.73 67.74 C-OEO8 3.56 68.24 0.0605 5.43 73.10 0-0612 7572 73.58 0.0458 7.87 77.83 0-0868 9377 78.62 0.0355 7.30 82.86= 0.0371 6f66 83.65 0.0242 8.56 87.881 300316 5.58 88.37 0.0182 9.59 92.98~ .mCPéB 6°25 93.36 0.0136 10000 <97-88 w.~307 6f52 98.35 0.0083 10.56 217 CA40(=.P')CA40* Ex: 7.670 MEV FF: 24.926 “EV EP- 30.044 MEv 440w") 007011 ERRSR ANWCM) 00700 ERRBR (0:3) (”3/SR) (x) (DEG) (MB/SR) (x) 12.37 5.3600 45°30 12.36 0°3150 40°00 17.52 0.3490 21510 17.50 0.2920 25°00 22.6? 3.3130 13330 2300“ 0.2780 10.00 27.30 2°2362 9°75 27°78 0°2560 6.20 32.62 0.2640 6°90 32°59 0°2100 9°00 37.75 0-2530 5'10 37°72 0'1850 6'40 42.27 3.2470 5°75 42.22 0.1910 4.40 48.30 3-2340 5°30 48°26 0°1840 5°30 53.40 3.2220 5°03 53.35 0°}720 #010 58.49 2’1920 4059 58044 001560 4.80 63.24 3.1660 4°75 63°20 0.1460 3°70 68.33 0.1560 4°43 68.27 0.1350 3.50 73.68 0.1530 4?20 73.23 0.1230 3.70 78.70 0.1670 3070 78067 001140 3050 83-75 0.1600 3°93 83.70 0.0990 4.40 88.47 0.1630 3°00 88-41 0°0913 3°50 93.47 0.1570 3300 93.42 0.0810 3°70 98.46 (.1430 3°00 98.41 0.9675 3.90 52: 34.775 42v 22. 39.828 MEV MSW") :0/30. L580? AP.G(CM) DO'IDO. ERRBR (era) (93/82) (Z) (050) (MB/SR) (X) 11083 C0309C 69903 12035 002860 40000 16099 3.2823 8590C 170k9 002750 21060 22.11 0.2680 17350 22.62 0.2440 16.67 26.94 0.2410 10:90 27.76 0.2160 7.38 320C7 001783 6:53 32058 001760 8018 37.23 0.1730 6°41 37.71 0.1640 5.00 42.31 0°1710 4°80 42.82 0.1570 1°20 47.41 0.1620 5300 48.22 0.1680 5.96 52.51 0.1580 4°30 53.31 0.1560 4.97 57.501 0.1550 4300 58.40 0.1450 5°36 62.6J7 0.1360 3°95 63.17 0.1250 5.45 67.74 :p1190 3°60 68.24 0.1050 4.53 73.1c\ 8-1050 6°10 73.58 0.0882 5.50 77.5H3 0.0905 6°30 78.62 0.0753 4-84 820,96 200357 5'65 83065 000596 4065 92.5v8 c.3557 4°20 93.36 0.0360 5.50 97.537 3.0469 4310 98.35 0.9294 4.85 218 CAhO(°:P')CA40* EX' 70865 MEV 9h 24.926 MEV EP- 30°?“4 ”EV A’ G(C”) 015/011 ERR0R ANGMM) DOIDQ ERRBR (DrS) (“R/SR) (x) (050) (MB/SR) (X) 12037 30602'0 159533 12036 00§920 16088 1.7053 305.450 11'37 17050 006030 12000 22.48 3.5130 4°46 22.65 0.5270 5.71 27.82 0.5380 4°86 27.79 0.8650 3°58 32.9: 0.5470 4.55 32.61 0.4730 5.43 37.7.? 3.4360 4°76 37.74 0.2920 4°18 42.90 3.3420 4°32 42.85 0.2120 4.36 ’1803’4 Q'ECBO .‘Jf97 48028 003.560 6000 53.43 “301540 5°87 53029 001230 5055 58.53 0.1120 7.00 58.47 0.1080 5.77 63.30 0.0814 6590 63.23 0.0606 6.16 68.37 3.0682 6°80 68.30 0.0537 6.07 73.73 3.0622 6°83 73.66 0.0503 6.42 72.78 3.0643 6°25 78.70 0.0469 7.34 83.30 0.0660 5°34 83.73 0.0454 7.86 88.51 0.0649 5°06 88.44 0.0386 6.08 93.51 0.0595 5°17 93.44 0.0357 6.10 98.50 0.0510 98.43 0.0303 6.17 531C {2: 34.775 vav EPs 39.828 MEv ARG(C“) 20720. ERRFQ AmG(CM> DGVDQ- ERRBR (DEG) (VB/SR) (Z) (020’ (MB/SR) (X) 11.23 3.7259 22.7C 12.35 0.6840 15.27 16.97 0.5750 12°35 17.49 0.6220 9.77 22.11 0.5370 9338 22.63 0.5600 5.33 26.94 0.5079 6°43 27.76 0.5350 3°40 32.0? 0.3580 4°28 32.60 0.3350 4.96 37.20 0.2860 4°64 37.69 0.1660 5.68 42.3? 0.1580 5°53 42.81 0.1340 5.44 4704? 0.1Q3C‘ 5‘37 “8023 001-260 7.48 52.53 0.1150 5363 53.32 0°1020 6°28 57°6fi? 3.3973 5°18 58.41 0.0947 5.89 62.7C) 0.0646 5.97 63.17 0.0730 6.29 6707‘ 20’351‘3 6'05 68024 000502 5079 73.12? 3.0409 13328 73":0 0'0365 8'88 78015~ C0039] 10°25 78064 0.0317 7'62 83.2c 0.0408 8°50 83.66 0.0308 6.66 22.231 0.0378 6§7C 88.37 0.9306 7.02 5x3.21 0.0309 5°84 93.37 0.0240 6.96 98.20 0.0285 532°Q 98.36 00Q180 6'33 219 CA40(D.P')CA4O* Exa 7-921 MEV (T‘ P= 24.926 WEV EP. 30.944 MEv A"~.13(C"") SCI/On. €389? ANG(CM) DO/DQ ERRBR (293) (Mg/SR) (z) (050) (MB/SR) (X) 12.37 0.2510 43.9; 12.36 0.2610 44.70 17.53 0.3100 18-75 17.50 0.3030 12.50 2?.69 0.3340 10-00 22.55 0.3350 7.62 F7.8? 0.3820 5594 27.79 0.3820 3.94 22.85 3.3840 5392 32061 0.3930 6.53 37.78. 3.362!) 4-24 37.74 0.3880 303‘} 42.90 0.3530 4384 42.85 0.3750 3.10 48.34 .3.3400 4694 48.28 0.3340 3.57 53.43 0.3180 3699 53.29 0.2920 3.22 58053 302760 329:7 5801+7 002320 3069 63030 0'2220 3'81 63.2 001980 3005 68.37 0.1880 3527 88.30 ooibao 3.09 73.73 0-1500 3-84 73-66 0.1250 3-85 78.78 0.1250 4834 78.70 0.0944 4.25 83.80 3.1080 4657 83.73 0.0696 5.50 88051 803944 4341 88044 000610 4040 93.51 0.0831 4650 93.44 0.0560 4.50 98.50 0.0780 4310 98.43 0.0518 4.51 EP 34.775 ”Ev EP- 39.828 MEV M.G(CM) 736/520. FQRoQ AMG(CN) DU/Dfl ERRGR (0&0) («a/SR) (Z) (DEG) (MB/SR) (x) 11.83 5.2820 55900 18.35 0-2810 35°33 16.97 C03380 20°53 17049 003380 16067 22.11 0-3910 11633 22.63 0.4150 6.61 26.94 0.4070 7639 27.76 o-fi430 3.76 32.09 0-4340 4351 32.80 0.3540 4-46 37.20 0.4040 3°92 37.89 0.4470 3.30 42.32 3.3640 3336 42.81 0.3830 3.03 47.43 0-3220 3-07 48.23 0.3290 3.73 52.53 0.2533 3606 53.32 0.2220 3.82 57.6? (~2070 3325 58.41 0.4540 4.16 62-70 nit-.1670 3043 63.17 0.1160 4.43 67.76- 5.1050 3344 88.24 0.0805 4.51 73012 (2.3723 (3.382 73060 000558 6089 78.16: 0.0594 5.75 78.54 0.0456 5.41 83.2%) 3.2485 5-85 83-66 0.0369 5.71 88021 000416 4368 88037 000285 6003 s£3.21 c.0387 4-93 93.37 0.9210 7.23 98020 (30333?1 “'0“? 98036 009185 6006 FP= ANG(C“) (DEG) 12°37 17058 22059 27.8? 32065 37078 42090 48034 53043 58053 63°30 68037 73.7? 78078 83080 88051 93051 98051 F9: A\G(CM) (DEG) 11.83 16097 22011 26094 32008 37020 42032 4704? 52-53 5706? 62.70 67076 7301? 78016 83020 88021 93021 98020 CA4O(°:P')CA40* 9409236 ”EV DC/Dfll (“Q/SP) 3.5450 305740 005953 C0630? 3.5420 0-4820 C0417O 3-3183 002700 O0PE4O C°1623 301280 001170 C-llec 0.1050 COC923 0.0857 300756 34.775 »” DOVDCL <83/SR> 003700 303670 303900 003840 503380 002800 302140 001460 0'1200 0'1020 0.0867 003756 303735 000703 300641 0.0597 303534 220 EQRBR (z) P1020 15303 7005 4366 4398 3344 3082 4374 4342 4090 4:82 4000 4093 4 4385 4°63 4329 4-12 4024 ERRQR (8) 40°03 22023 10090 6391 4:64 4356 4385 5002 4°49 4085 4°81 4334 6073 7017 7339 3771 4-06 3082 Ex8 80097 EPI 300944 MEV ANG(CM) (DEG) 12036 17050 22065 27079 32061 37074 42085 48028 53029 58047 63023 68030 73066 78070 83073 88044 92044 98043 EPs 39.§28 MEV ANG(CM) (DEG) 12035 17049 22063 27076 33060 37'69 42081 48023 53032 58041 63017 68024 73060 78064 83066 88037 93038 98037 MEV DOVDfl (MB/SR) 004950 004850 004900 005060 004850 003720 002850 002180 001760 001420 001170 000930 000820 000756 000750 000732 000676 000610 DGVDII (MB/SR) 004560 004500 004780 004480 003140 002460 001730 001380 001280 001130 000954 000783 000682 000645 000585 000488 000400 009304 ERROR (X) 18075 22000 8000 3076 5070 3068 3077 4080 4064 5033 4034 4010 5013 5026 5067 4020 4073 4020 ERRBR (X) 27090 16042 6078 3068 5051 4084 5023 6053 5004 5015 4075 4078 6042 4096 V 4050 5059 5017 4083 EP= AhG(CV) (EEG) 1203‘ 1705? 22067 27081 32064 37077 42089 4303? 5304? 5806? 630?? 68036 7307? 78076 83079 88040 93040 98040 CAQO(°;P')CA40* 240936 50700; (”3/SR) 301320 0.3513 Q'P940 3'3070 3.3123 C'BQC? 3-9930 3-2573 202360 '2360 .1973 001530 301370 p01920 01093 01040 .3960 03983 ) (J I__) OL)( x.) L) 002380 0'1830 301480 301160 803993 3.3934 :0Q952 303785 600651 221 ERHBQ (%) 5138: 21°60 10'60 6057 7010 4353 4396 5048 #375 4036 5920 4523 4°00 4039 4905 #326 4326 3957 EX8 8.361 EPI 300044 MEV ANG(CM) (DEG) 12036 17050 28066 27079 32061 37074 42085 48028 53038 58047 63023 68030 73066 78070 83073 88044 93045 98044 EP= 39.023 MEV ANG(CM) (pas) 12035 17050 22063 27077 32058 37070 42082 48024 53033 58042 63012 68025 73061 78055 83068 88039 93039 98038 MEV D0/ D9. (MB/SR) 002540 002880 003170 003470 003560 003360 003420 003030 002640 002150 001720 001550 001250 001050 001020 000936 000858 000774 DU/DD. (MB/SR) 003150 003670 004220 004350 004480 004350 004130 003820 003240 002260 001710 001260 001040 000855 000772 000723 000563 0.9357 gRRBR (X) 32030 15070 8092 4002 4054 3092 3'16 3064 3045 3085 3041 3013 4003 3090 4034 3070 3060 3058 ERRBR (X) 30067 16053 6054 3000 4043 3030 2087 3093 3043 3039 5000 4006 5011 4064 3081 4039 4017 4032 EP ANG(C“) (DEG) 12036 17052 22°67 ?7031 3206“ 37077 42039 4803? 5305? 5805? 63029 $8035 7307? 78074 83079 E8043 9304? 9804C ED AhStC”) (DEG) 11-3“ 16098 22012 26095 32039 37021 42033 47044 52.54 57063 62070 67077 73-13 78017 83-FO E802? 9302? 98021 CA40(9;P')CA#O* 240926 “EV 3070.0. (”5/SR) oZSOc 202890 .3910 .3450 03340 03P7C '3C5O 002540 5°220C 002020 30187C 3.1340 001623 301573 0.1350 C0195? $00970 \- 3-3753 .3 A (W L") (3 ( .gga 34-775 N SUVQQ. (HQ/SR) 3.2940 003470 003660 303830 203740 222 ERRBR <2) P9315 15f55 5909 6f68 6'53 4316 5306 8°40 8°60 8'40 708: 3392 8010 6'80 3392 3-43 6343 6350 EQHQR <%> 5505C 21330 12f70 7f35 4321 6-30 6.50 6§55 7960 7'50 7f80 7-70 6‘15 12360 12990 7370 8°53 8°03 Ex- 0.412 EP3 300944 MEV ANG¢CM) (pas) 12036 17050 22065 27079 32061 37074 #2085 48028 53038 58047 63023 68030 73066 78.70 83073 880#4 930Q5 98044 EP- 39.§28 MEV ANG(CM) (DEG) 12035 17050 22063 27077 32058 37070 42092 48024 53033 58002 63012 68025 73061 78065 83058 88039 93039 98038 MEV D0709. (MB/SR) 0-2700 003150 003080 003520 003530 003460 003060 002450 001950 001540 001380 001240 001090 000960 000886 000680 000610 0.9430 06/ DD. (MB/SR) 003350 003620 009020 004170 003940 003430 002580 001830 001250 001030 000920 000770 000660 000520 000430 000358 000270 0.9180 ERRBR (X) 35071 17085 8022 “043 4078 3080 3042 8000 7070 8000 7070 6020 4033 3078 “031 6080 9070 9050 ERROR (1) 27050 14000 6067 3012 4082 6090 6060 10000 8040 8060 9060 4070 6000 5055 8050 8040 8040 80k0 “‘- _’ _ “813w ' U i 223 CA40(D:P')CA40* EX3 80557 MEV F9= 94-9?6 vEV EP' 3009“4 MEV Maw”) 0070:). ERRS‘? ANG(CM> DU/DD- ERRBR (0:0) (Ma/ea) (z) (050) (MB/SR) (x) 12.39 0.4980 26020 12.37 0.5030 15-90 17.54 0.4200 18?32 17.51 0.4020 13.40 22°69 2.3780 99 5 22.66 0-3500 8078 27.8? 0.3950 6-20 27.80 0.3420 4031 32.66 :-3500 6f55 32-62 0-3300 6'25 37.70 3.3440 4977 37°75 0-2560 “'51 42.9 5.2840 5017 42.g6 0.2160 4.85 48.34 0.2520 567? 48.29 0o2020 4.96 53.44 0.2320 Sé13 53.39 0-1900 4'85 58054 202092 5'11 58048 001850 4.07 63.31 0.1720 5634 63.24 0.1570 4.15 £8.3S 0.1440 4.55 58.31 0.1270 3-81 73.74 9.1287 4677 73.67 0.0954 4.73 7807Q 3'1173 4‘64 78071 009752 5026 53081 301173 4367 8307‘? 000640 5068 88.52 0.1160 3.95 88-45 0.0530 5°65 93.5? 0.0990 3°91 93.46 0.0460 5.22 98-5? C0396“ 4f06 98045 009410 4096 PP 14.775 ME‘v EP- 39.828 MEV .a0(c”) 90700. £880? AkstCM’ DCVQQ- ERRBR (Ere) (VB/SR) (z) (DEG) (MB/SR) (z) 110?“ 304539 36f03 12035 004750 18066 1e.92 0.3920 17320 17.50 0.4206 14.67 22.1? 0.3540 12-33 22.63 0.3820 10.27 26.95 0.3660 8-69 27.77 0.3306 4.89 32-0° 0-2910 5322 32-58 0-2787 5-85 37.21 0.2410 5°93 37.70 0-2161 5-47 42.33 Q.209C 4396 42.82 001823 5027 47.44 0.1920 4753 48-24 001860 5'77 52.54 3.1850 4-10 53.33 0.1888 4.22 57.63 0.1700 3389 58.42 0.i565 4.37 62.70 9.1450 3?95 63.18 0-1220 4035 67.77 3.1140 335: 68.25 0.0954 4.26 73.13 0.0991 7005 73061 0.0612 6.74 78.17 0.0651 8§15 78.65 0.0460 7.25 83.20 0.0505 8047 83.68 0.0382 5-75 57.9? 0-c429 4-84 88-39 0'9395 7'4“ 92-3? D-CESG 5°53 93-39 0-0240 6058 97091 303997 5533 98038 009180 5098 A43(CK) (055) n \ :84C(”: {1‘ 224 P')CA43* 1311'") 'J k“ Al','\. .J x.) I :) c) c) c) a) L) L) u) L) L) ( ) D O O 2') f) ('3 J17 (Iv 13-0? 14°03 1 . . f‘ 7" -2 00 Q0hW " «J V 10°”? 0..) 5-; ...-D 1‘) I‘.) I‘.‘ V" LI {7! X II EP= 300944 ”EV A\G(:V) (0E3) 17051 22056 P7081 32053 37.76 42057 48030 53040 58049 630?5 63032 730b8 78072 33075 88045 930%6 98- 0’05 EP ,xG(C”) (DEB) 17.50 22053 27078 32059 37071 420£3 48.35 53034 58042 63019 680?6 73062 78.08 830$? 58040 9304C 98039 R0743 MEV DU/DQ. (MB/SR) 002350 002150 00;930 001220 000788 000692 000643 000640 000606 000448 000365 000292 000270 000230 000250 000170 0.0112 39.828 MEV DO/MQ (MB/SQ) ‘002250 001950 001730 004290 000670 000456 000430 000460 000417 000330 000220 000146 0.012# 000125 009112 000088 000065 ERRBR (X) 40000 15000 8000 13000 11000 11000 12000 11000 10000 9000 8000 11000 12000 13000 9000 11000 12000 ERQBR (X) 21°00 15000 8020 11000 15000 14000 15000 12000 11000 12000 8000 15000 20000 '11000 12000 11030 18000 Iiflfi‘“ 22$ CA40(D.P')CA40* Ex8 3-270 MEV pp: 34.9?6 ”EV EP8 30094# MEV mam“) 907m FRRQQ Anew”) DU/Dfl. ERReR (DFG) (NS/SR) (z; (935) (MB/SR) (x) 2708? (.PEQQ 13003 27079 001830 11000 32065 3.1589 1200: 37074 000870 10000 3707? C'lFZC 6'03 42085 000720 9000 «2.99 c.1053 10.53 48.28 0.0560 12.00 48.34 3-364C 15033 53.38 0.0410 13.00 53.03 e-cnac p330; 63.23 0.6310 11.00 58.63 3.537: 1530: 78.70 0.0207 12.00 73.73 3.3352 12303 83070 000138 12000 88.44 0.9126 11000 SP 34.770 ‘ HP: 39.820 MEV «ue (x) ?708? 00064C 35000 32065 c.066C 25.00 4303# 000720 12.00 33.43 0.0750 12.00 bF-03 0.073: 12.00 57030 0.072C 10.00 f:'37 003690 10.00 £5026 0.0600 9.00 25.01 0.0530 10.00 &~.§3 0.0500 9.00 ;f0-. 000480 8000 p*051 0.0460 8.00 AHG(C“) (6E1?) 270?“ 32.0% 42039 47.43 52053 69.7? 7301? 73015 9702? g7oq1 9?091 9709“ A4C(D:p')CA43* 007 00. (WU/SR) QOC3IC 0.0325 0.0350 000390 0.0440 000462 000386 000305 0'026“ 0.0220 0.0196 00016? 0-0120 Ex: EP=340775 MEV MEV ERRBR ‘(x) 17.00 17.00 17.00 14.00 11.00 11.00 11.00 16.00 17.00 17.00 11.00 12.00 11.00 _223 CA40(D.P')CA4:. Ex: 3.974 MEV EP834-775 MEV A23(C“) 00709. ERRUR (0E9) (MB/SR) (%) 26.09 0.0494 11.90 32.01 0.0537 16.80 37-1? 0.0565 15.50 «2023 0-0580 12.90 47.33 0-0633 8-70 52-#2 0.0622 9.50 57.50 0-0640 6.90 62057 000563 6.50 57063 0.0540 5090 72.63 0.0448 5.30 7900? 000355 11.30 a3-04 0.0287 11.50 87.76 0.0185 7080 92.75 0.0147 9.90 97075 000116 9.70 CA40(D.P')CA40* i) (0E0) 27.71 32.52 37.63 42074 48.14 5302? 58031 63007 68.13 73043 79.52 33054 53075 930?6 9Q.?5 \G(C%) Ex: MEV Ep3390828, MEV DGVDQL (YB/SR) 0.0451 000502 0.0510 3.053? CvOSaS 0'0583 000566 000456 0-0409 000310 000225 000151 000129 000096 000090 ERRCR (x? 14.90 18.70 22.20 24000 13.70 10.00 7.70 10.00 7.10 9.90 7.00 12.70 14.00 12.80 9.50 CA4O(D.P')CA4C* a23CA43* Ex= 9-856 WEV ‘ EP=34.775 MEV my) 3070!). ERRUR (9E0) (vs/SR) (x) 26.93 0.0877 12.70 32.32 c-O954 10.50 37.13 0.1013 10-50 43.?h 3.1063 8.10 47.34 0-1100 9-40 59.43 0.0987 6.30 57-51 3.0945 5.70 53.59 3.0753 6.10 57-65 o-oeaé 5.60 73-6: :«3434 10.20 78-04 9-3335 13.30 $3006 CoCP65 13.70 47.78 Q.C225 8.00 99.79 3.0152 10.40 97.77 0.0140 9.00 CA40(°ID')CA#O* 1‘..\\; l3 ( CK’) (0E‘5) 26.93 32.02 37014 42°24 47.34 52.43 57.52 52.59 67.65 72070 79.04 83007 87.78 92078 97.77 CA40(°ID')CA40* A533(C‘~4) (0E5) 22.13 26090 32002 3701“ 42025 47035 5204# 57.52 62.59 67.66 72.71 73.05 83°07 87.79 92.79 97.73 235 EX=100045 EP=34-775 2‘6 / DD. (MB/SR) 0.1080 001160 0.1185 0'1230 c.1140 001100 0.0985 0.08#0 0'06?!+ 000395 0.0345 0.0210 0'0195 0.0168 0.0128 EX'130276 EP‘340775 30700. (MR/SR) 0.1300 001497 09158“ 001590 0.1494 0'1274 0.0872 0'0662 0.0478 0.0410 0.0327 0.0327 0.0280 0'0250 0-0202 000175 MEV MEV ERRHR (X) 19.40 10.00 9.70 7.00 10.50 10.30 5.90 6.20 6.00 6.00 12.60 15.60 8.50 10.10 8.20 MEV MEV ERRQR (X) 20.00 19.00 7.30 8.§0 7.00 7.?0 7.00 7.90 8.30 8.00 8.00 14.10 10.90 7.30 8.50 8.30 -F1 IES um ”'7HifixfigflnfiifijfiuWWW