more spm rue IN THE REACTIONS 12mm) 120 (4.44) AN01208n(p,p’ >120-s'rfi 1.17) Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY JAMES J. KOLATA 1969 {Uses This is to certify that the thesis entitled PROTON SPIN FLIP IN THE REACTIONS l " 2OSn(p,p')1203n’(1.17) 't 12C(p,p')12C:(u.uu) AND presented by James J. Kolata has been accepted towards fulfillment of the requirements for ' Ph.D. degree in Physics @flwfifiaflwzx Major professor Date February 11+, 1969 0-169 {4:311 4‘ ~ ABSTRACT PROTON SPIN FLIP IN THE REACTIONS 12 3': 3': C(p,p')l2C (u.uu) AND l208n(p,p'>1208n (1.17) By James J. Kolata scattered from the first 2+ state in 12C and 12OSn has been measured at incident proton energies of 26.2 and no.0 MeV 12 for C, and 30.0 MeV for l2OSn. The experimental method along the normal to the scattering plane. It can be shown that this correlation is directly proportional to the spin— Angular distributions were obtained over an angular 12 range of 25° to 155° in the laboratory system fOP the C target, and from 30° to 155° for the 120Sn target. The data display prominent backward peaks similar to previous Observations at lower energies and for other nuclei. The magnitude of this peak in the spin-flip probability was about 0.30 for 12c and 0.50 for 1205n and the location of its rapidly rising edge seems to be correlated with the James J. Kolata target mass number. The total spin-flip probability is only 2C and 0.08 for l2OSn despite the large about 0.03 for backward peak in the angular distribution, because the inelastic cross section is largest at the forward angles where the spin— Distorted— flip probability is small.(};CL10 fbr both targets). wave calculations were performed with collective—model and microsc0pic—model form factors in an attempt to determine the type of information about spin—dependent nucleon-nucleus forces which can be extracted from spin-flip data. The theoretical predictions were in semi-quantitative agreement with experiment The most serious failure in at the peak of the distribution. 12C data at no.0 MeV where the this regard occurred for the predicted peak spin—flip probability was 0.20 compared to the Larger differences were observed measured value of about 0.30. 1 In the case of 2C, these for the forward angle data. discrepancies were associated with the failure of the optical model for this light nucleus, and no definite conclusions could be reached regarding the spin—dependent part of the inelastic interaction. For the l2OSn data, however, there is some evidence that the observed discrepancies are related to the spin—dependence of the inelastic interaction. If this is the case, a more adequate treatment of this interaction may significantly improve the agreement between theory and experiment. PROTON SPIN FLIP IN THE REACTIONS 12 120 '6 1‘ 12C(p,p')12C’(u.uu) AND OSnCp,p') Sn’(1.17) BY James J. Kolata A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1969 ACKNOWLEDGMENTS I wish to express my deepest appreciation to my thesis adviser, Dr. Aaron Galonsky, for his guidance and continuing help and encouragement throughout the course of this work. I am grateful to Dr. Barry Freedom for his invaluable assistance in the Optical—model analysis, and for a careful and critical reading of this paper. To the members of the Cyclotron Laboratory technical staff, and especially to Mr. Norval Mercer, go my thanks for their assistance in the design and construction of much of the apparatus used in the course of this experiment. Special thanks go to Richard Howell and Ronald Sager for their help in taking the data, and to Dr. Hugh McManus and the members of the professional staff of the Cyclotron Laboratory for their helpful advice and assistance. I also acknowledge the financial support of the National Science Foundation for the experimental program, and for the personal support provided by a two—year Co—operative Graduate Fellowship. Very special thanks go to my wife, Ann, for her encouragement, understanding, and support during the past year, and for typing this thesis. ii Acknowledgements List of Tables List of Figures Chapter I. Chapter II. II.A. II.A.1. II.A.2. II.A.3. II.B. II.B.l. II.B.Z. II.B.3. II.C. II.C.l. II.C.2. II.C.3. Chapter III. III.A. III.A.l. III.A.2. III.A.3. TABLE OF CONTENTS INTRODUCTION DWBA THEORY OF ANGULAR CORRELATIONS Transition Amplitude General Form of the Transition Amplitude for Inelastic Scattering Zero—Range Approximation Reduced Amplitudes Cross Sections, Polarization, and Angular Correlation Scattering Cross Section, and the Polarization of the Scattered Particles Statistical Tensors Angular Correlation Function and Spin Flip Nuclear Reaction Models Collective Model Microscopic Models Nuclear Wave Functions EXPERIMENTAL METHOD Beam Line Cyclotron and Beam Analysis System Beam Alignment Targets iii ii vi vii 12 13 15 19 22 25 25 25 27 28 Chapter Chapter III. III. III. III. III. III. III.C III.C III.C .C. D D III III. III. III.D III.D .D. D D III III. III. IV. IV. IV. IV. IV. V. P E” P >>E> s &> O ?3E° w U3t> > 2 3 u. 5. 6. Target Chamber Charge Integration Detectors Gamma—ray Detector Assembly l2C Experiment Proton Detector for the Proton Detectors for the 120Sn Experiment Electronics Fast Timing Circuitry Side-Channel Amplitude Analysis Data Collection Circuitry Data Reduction Angular Correlation Function and Spin—Flip Probability Gamma Detector Efficiency Measurements Acceptance-Angle Corrections Accidental Coincidences Pulse Pileup and Dead Time Losses Analysis of Experimental Uncertainties OPTICAL MODEL ANALYSIS Elastic Scattering Wave Functions and the Optical Model The Search Procedure 28 29 30 30 33 38 90 43 45 53 5M 5M 58 63 65 69 75 78 78 81 Optical—Model Parameters for 12C(p,p)120 83 Optical—Model Parameters for 120 120 Sn(p,p) Sn EXPERIMENTAL RESULTS AND COMPARISON TO THEORY. l2c Differential Cross Sections Inelastic Asymmetries Spin-Flip Probability 1203n(p,p')1205n* (1.17) Differential Cross Sections Spin Flip iv 90 9# 94 94 98 101 107 107 111 Chapter VI. SUMMARY AND CONCLUSIONS 118 Appendix A. THE BOHR THEOREM 121 Appendix B. ACCEPTANCE—ANGLE CORRECTION 125 Appendix C. DERIVATION OF THE FORM FACTOR IN THE QUASI-PARTICLE MODEL 133 References 137 Table III-1. III-2. IV-1. IV-2. LIST OF TABLES Measured values of the efficiency of the gamma— ray detector for the 12C and 120Sn experiments. Measured values for the strength of the 60Co source . Optical-model parameters which produced the fits to the 12C elastic data shown in Figures IV—l to IV—3. Optical—model parameters for elastic proton scattering from 120Sn at 30 MeV. Entrance channel optical-model parameters used in the DWBA calculations. Exit channel optical—model parameters used in the DWBA calculations. Total spin-flip probability. Comparison of the B(E2) values obtained from proton inelastic scattering to the results of gamma—ray measurements and to the theoretical predictions. Average of direct and interference terms over the acceptance angle of the gamma-ray and proton detectors. vi 60 60 85 92 95 96 102 112 132 Figure III-1. III-2. III—3. III-4. III-5. III-6. III—7. III—8. III-9. III-10. III-ll. LIST OF FIGURES Experimental area of the Michigan State University Cyclotron Laboratory. Circuit diagram of the ORTEC Model 265 photomultiplier base (printed by permission of ORTEC, Inc.; Oak Ridge, Tennessee). Diagram of a zener—diode voltage regulator for the last four dynodes of the photomultiplier. Cross section of proton detector package. 12 . Typical proton spectrum for C experiment. 120 . Typical proton spectrum for Sn experiment. Block diagram of the coincidence circuit. Schematic representation of leading—edge and conventional crossover timing, and the definition of 'walk' and 'jitter'. Typical time Spectrum obtained with the circuit of Figure III—7 for the 511 keV gamma—rays from 22Na. Typical time spectrum obtained for the gamma—ray cascade in 60Co (E,.= 1.17 and 1.33 MeV). 12 Typical time spectrum for the C experiment. The width of the central ('TRUE+CHANCE') peak is 1.1 nsec (FWHM). The remaining ('CHANCE') peaks are broader due to a contribution from the beam PUlse width. Resolving time after subtraction of chance events is 0.9 nsec (FWHM). vii 26 32 3M 35 39 41 H2 nu H6 H7 H8 III—l2. III-13. III—1H. III-15. III-16. III-l7. III-18. IV-1. IV-2. IV-3. Typical time spectrum for the 120Sn experiment. The exponentially decaying 'tail' is due to the decay of the 5.5 nsec state at 2.28 MeV (see Figure III-13), which cascades through the 1.17 MeV state. H9 Energy level diagram for the low-lying excited states of 120Sn. 50 Effect of the slow-coincidence requirement on the time spectrum for the 12C experiment. 52 Quadrupole radiation patterns for pure (l,m) multipoles. These are polar plots of intensity vs. angle relative to the axis of quantization (z-axis). 57 Gamma—ray Spectrum for the 12C experiment, showing a typical setting of the pulse—height window. The Compton—scattering continuum extends to zero pulse- height below the lower edge of the window. 59 Decay scheme for 60Co. 62 120 . Gamma-ray spectrum for the Sn experiment, showing the exponential distribution assumed in the pileup calculations (Sec.III.D.5). 71 12C elastic cross section and polarization fits obtained for the 26.2 MeV data (Di 63,Cr 66) with the optical—model parameters of Table IV—l. Cross sections shown in ratio to Rutherford . 86 scattering. 12C elastic cross section and polarization fits obtained for the H0.0 MeV data (B1 66a) with the - 87 Optical—model parameters of Table IV 1. 12 and polarization fits C elastic cross section Obtained for the H9.5 MeV data (Fa 67,Cr 66a) with the optical—model parameters of Table IV—l. 88 viii IV-H. IV-5. IV-6. V—l. V~2. Energy dependence of 12C Optical—model parameters. 'Error bars' indicate the range over which the parameter can vary with less than 25% change in 2 XT' 89 Typical 'map' of X2 space for the spin—orbit diffuseness parameter. 91 120 Sn elastic cross section and polarization fits obtained for the 30 MeV data (Ri 6H,Cr 6H) with parameter set 2C of Table IV—2. This fit is typical of those obtained for 120Sn. 93 DWBA fits to the 12 at Ep=26.2 MeV (Di 63) and H0.0 MeV (Bl 66a). The optical-model parameters are listed in Tables V—l C inelastic cross section data and Vn2. 97 DWBA fits to the 12C spin flip, and to the inelastic asymmetry data (Cr 66), at Ep=26.2 MeV. 99 DWBA fits to the 12C spin flip, and to the inelastic asymmetry data (Bl 66a), at Ep=H0.0 MeV. 100 Dependence of the spin—flip and inelastic asymmetry predictions on the spin-orbit optical parameters. The notation (':2S%') refers to the upper and lower limits for the parameters listed in Tables V—l and V—2. 105 Dependence of the spin-flip and inelastic asymmetry predictions on the spin—orbit optical parameters, and on the 8:1 part of the microscopic-model interaction. 106 Collectivermodel DWBA fits to the 12OSn inelastic cross section data (Ri 6Ha) at 30 MeV. The identification numbers 18 and 2C refer to the optical-model parameter sets of Table IV-2. The deformation parameter is also given. 108 ix 120Sn inelastic Microscopic—model DWBA fits to the cross section data, using impulse-approximation form factors. 109 12OSn inelastic Microscopic—model DWBA fits to the cross section data, using Kallio—Kolltveit form factors. 110 Collective—model DWBA fits to the 12OSn spin—flip data. 113 Microscopic—model DWBA fits to the 120Sn spin- flip data, using impulse—approximation form factors. 11H Microscopic—model DWBA fits to the 120Sn spin— flip data, using Kallio—Kolltveit form factors. 115 Composition of a reflection in the scattering plane (x—y plane) by a rotation of 180° about the z—axis followed by a parity inversion. 122 Detector geometry for the calculation of the acceptance—angle correction. 129 I CHAPTER I INTRODUCTION Several experimental techniques are available for studying the spin dependence of the nucleon—nucleus reaction. In particular, one might investigate the inelastic scattering of polarized protons (Fr 67,G1 67) or the effects of target polarization on a given reaction (Go 62). Either of these methods involves the preparation of an initial system with known spin orientation; the relative scarcity of such data reflects the experimental difficulties encountered. Alternatively, it is possible to determine the angular dependence of polarization of the residual nucleus, when the initial system is completely unpolarized. Usually, one observes the angular correlation involving the scattered particle and the de—excitation gamma radiation. It can be shown (Go 62,Sa 6H), in the context of the distorted—wave Born approximation (DWBA) with unique total transferred angular momentum, that the information obtained by this method is the same as that obtained by scattering from polarized targets. Therefore, such measurements can provide valuable data concerning the spin dependence of nuclear reaction mechanisms for the wide range of nuclei for which polarized targets are unavailable (as, for example, 1 2 if the nucleus to be studied has zero ground state spin). In addition, they can provide supplementary information in those cases for which the inelastic scattering of polarized protons has been measured. The chief disadvantage of the method lies in the need to perform a coincidence experiment. The angular correlation function for the case in which gamma radiation is detected in the plane determined by the incident beam and the scattered particle (in—plane correlation) has been analyzed in the DWBA by several authors (Sa 55,B1 61, Ba 57). Banerjee and Levinson (Ba 57) predicted the form: We )- A+ 393209 -e I Ire-#69 '6) (1'1) 7 ' T l 1‘ 1 and associated the last term with the presence of spin flip in the interaction. Such a term has been observed (Yo 60, Br 61), but it has proved to be very difficult to extract the relevant spin—flip probability, which is expected to be quite sensitive to the spin dependence of the nucleon—nucleus interaction. Recently, Schmidt et.a1. (Sc 6H) have pointed out that spin flip could be more easily studied through an angular correlation in which the gamma radiation is detected along the normal to the scattering plane (gamma-perpendicular correlation). They were able to show that this correlation is directly proportional to the spinnflip probability for the case of a0+ — 2+ transition, independent of the reaction .mechanism assumed. The argument may be extended with minor .modification to the excitation of a 1: or 2— state from a + 0 _ground state. We have used this method to investigate proton spin flip in the excitation of the first 2+ state of 120 and 120 Sn. These targets were chosen for several reasons. First of all, 12C is a nominally 'closed shell' nucleus on which numerous theoretical calculations have been done. In addition, the relatively small number of open reaction channels and the well—separated energy levels are non—trivial experimental advantages. Unfortunately, 12C is also a notoriously poor 'optical—model' nucleus, in that it is extremely difficult to extract optical model parameters which accurately describe the elastic scattering (Sa 67). When it became clear that good optical parameters were necessary for the interpretation 12OSn which is of the data, it was decided to investigate not plagued by this problem, though it is experimentally more difficult because of the high background of gamma radiation from various reactions in the target. The data have been analyzed in the DWBA, with several different reaction models, in an attempt to determine the type of information about spin-dependent nucleon-nucleus forces which can be extracted from spin~flip measurements. Chapter II is devoted to a discussion of the DWBA method, and of the particular reaction models used. The experimental method and techniques are described in Chapter III, followed by a short discussion of the Optical model and the extraction of the parameters in Chapter IV. The theoretical predictions are compared to the experimental data in Chapter V, and the resulting conclusions which can be reached from this comparison regarding the value of spin—flip measurements in the investigation of spin—dependent forces are summarized in Chapter VI. ax. CHAPTER II DWBA THEORY OF ANGULAR CORRELATIONS The theoretical formalism of the distorted—wave Born approximation (DWBA) for inelastic scattering has been treated in detail by Satchler (Sa 6H) and by Tobocman (To 61) The basic assumption made in the development of the theory is that of 'weak coupling'; that is, it is assumed that elastic scattering is the most important process that occurs, and that the inelastic event can be treated as resulting from a perturbation which causes transitions between elastic scattering states. The elastic scattering itself is treated 'exactly', in the sense that it is calculated from an optical model potential using parameters which fit the elastic data. In the following sections, some of the more important results in the development of the DWBA theory, and its application to the prediction of angular correlations, are discussed. The treatment followed is that due to Satchler (Sa 6H). II.A. Transition Amplitude II.A.l. General Form of the Transition Amplitude for Inelastic Scattering In the DWBA theory of inelastic scattering, the transition amplitude takes the form: 74H " I 5— H 7’. (1&3) (HIV/71> gmfxefq.) 3'; 4'; (11.1) "t”? '1'? 5 ‘ I I "i "I where P; and F; 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 coordinates. The functions Ct“ and it; are the distorted waves, lo which are eigenstates of elastic scattering from the target states, respectively. They are in its initial and final usually generated from an optical model potential using parameters which fit the elastic scattering data (see Chapter IV). The superscript (+) or (-) denotes outgoing or incoming boundary conditions, and the subscript m refers to the z—component of the projectile spin. The two boundary conditions are related by time reversal invariance: I m-m (4') a A (- r 22%;»: 3'” ) (11.2) (are (-) Since the spinaorbit term in the usual Optical model potential (see Chapter IV) can couple different spin projections, the distorted waves are, in general, non-diagonal matrices in (I) 'f 9' ~- ~. ‘v (I) ‘v- spin space. The offediagonal terms (mim') can lead to a nonzero spin—flip amplitude. The remaining factor in the expression for the transition amplitude is the matrix element of the interaction causing the transition, taken between the initial and final internal states of the scattering system. It contains all of the information about the structure of these states and the mechanism which couples them, and can be looked upon as producing transitions between the elastic scattering eigenstates Z" and z; Since this matrix element will, in general, be spin dependent, it can also couple different spin projections and therefore produce a nonzero spin—flip amplitude. II.A.2. Zero—Range Approximation The general form of the transition amplitude (II.1) involves a six dimensional integration over the space of T2 and'PE. Since the numerical evaluation of such an integral is difficult and time consuming, the 'zero-range' approximation is usually introduced. The physical assumption behind this approximation is that the scattered particle is emitted at the same point at which the incident particle is absorbed, so that'Fg = (§)'?' (where A and B are the masses Of the target nucleus in the entrance and exit channel). The introduction of the zero—range approximation reduces the transition amplitude to a threeudimensional integral which is much easier to compute. The price paid for this simplification is that the effects of particle exchange are neglected, and possible nonlocal inelastic interaction potentials cannot be introduced exactly. However, both these cases can be treated in some approximation by replacing the interaction potential by an equivalent local but momentum dependent pseudo~potential (Pe 6H). II.A.3. Reduced Amplitudes The transition amplitude Tfi is usually expanded in terms of 'reduced amplitudes' corresponding to the transfer of a definite total angular momentum j, orbital angular A momentum l, and spin angular momentum E'to the nucleus during the inelastic event. In the zero~range approximation, this expansion takes the form: T z (Z'I)&< 'M m «I ”a"; - = J'i' JEJ ' " ' I“ f; ‘51, l. ’ f c f f) 35.] (I) (II.3) Main" -m., and J and M are the total angular :: _.+ where m Mf Ml mf The momentum of the target nucleus, and its z—component. ”'5'"; . expression for the reduced amplitude g”. in terms of previously defined quantities appears in Ref. (Sa 6H). )a-f The transferred angular momenta (1,8,3) are determined from the relationships: A A A A 3": If-Ji ‘5' = 51%} l = j-8 (II.H) and, in the zero~range approximation: 72}1fl;.:= (“9“ x 73’ .I- .s . where the transition is (J )i“’ (J )f and Si(sf) 13 the spin of the incident (scattered) particle. It is important to notice that the value of each of these angular momenta during the inelastic event is to be used; this is not necessarily the same as the asymptotic value. For example, a reduced amplitude labeled by s=0 may still contribute to spin flip (s=l asymptotically) through the distortions introduced into Czi and CK; by the spin-orbit term in the I optical potential. II.B. Cross Sections, Polarization, and Angular Correlation II.B.l. ScatteringACross Section, and the Polarization of the Scattered Particles The differential cross section for an unpolarized projectile and target is proportional to the square of the transition amplitude and can be written in terms of the reduced amplitudes as: 40' «us/‘4 in J’ I WWI-2 ”(9) - ---—'b “ .944- Z_ IEP. {(11.5) J 15 l aIn " (2MB? 4'}: Ewes») SJ where ,AQL and “‘96 are the reduced masses in the entrance and exit channels. Note that the sum on j is incoherent, although the possibility of interference terms between different 1 and 5 remains. The vector polarization of the scattered particle is defined as the expectation value (SID/Sf. . If the z—axis . A , .A A is chosen along ka and the y—ax1s along ka x kb’ the “Y e“ {‘r- 1” AA ‘4 Ilr 10 expression for the polarization is: is M DI‘. "l: “N" M‘il’M‘ * {[(s -m )(sun ")1 g , , ., ) W9): — ‘ g 4: ‘ ”IF“, (fit-H '1 (II.6) ”WW "me": * s, 2 P153 (F425- ) where the sums are taken over l,l',s,s',j, and all the projection quantum numbers. Here again the coherent sum on 1 and s appears. laboratory. In the final version, an option allowing the COherent sum over 1 and s was implemented, and the expression (II.6) for the polarization was programmed. In addition, a subroutine to calculate the spin flip was added; this Particular routine will be briefly described below. II.B.2. Statistical Tensors It is convenient to describe the polarization of the reSidual nucleus in terms of the density matrix (Br 62) in Mf for the residual nuclear spin, which is constructed from the reaction amplitudes: 7: Ti£("‘.‘mi"'6 Mp) T:‘.(m‘-Mi..ffl\,’) (IL?) l. - ~ 'Mk huq‘NI & This in turn.may be expanded in polarization moments or 'statistical tensors' KCQ of rank K 5 2Jf ff. _ TM.” {Mf’ " if-) ’:q (II.8) After a moderate amount of algebra, we find for the statistical tensors: 3"..J'£ +5 +JI-K4-Q-A gm ’0 34 H) ‘52". (-) WZJM) (ail-7) WG-5 3,1,3)!“ ‘1’:le " '7‘ "i (11.9) M'Q’ "F "‘- * HM It‘- X (Lisa-’05“ ’ KQ) F135 ‘9 (50‘s.: ) Ifluntamc= HHHE—HE” In contrast to the expressions for the differential cross-section and polarization, the sum on j is Coherent, so that amplitudes with different total transferred angular momenta can interfere. The flcq are constructed so as to behave under rotations like spherical tensors of rank K. In addition, when referred ch .- -L . - tO ka as the z—axis and ka x kb as the y-ax1s, they satisfy the symmetry relation: ('0 79m = ('9 fit -0 (11.10) J SO that “the ac are real (imaginary) for K even (Odd). In Particular, IOKO . vanishes for K odd. The Spin—flip subroutine calculates the fit? for unique 12 total transferred angular momentum j 5 2 and the possibility of coherent sums on 1 and s is retained. Thus, it is sufficiently general for inelastic proton scattering to a 2+ level from a 0+ ground state, where the only allowed values of the transfer quantum numbers are (l,s,j) = (202) or (212) (see II.H above). The statistical tensors are computed for all K,Q satisfying: 051(5 2J (11.11) —K 5 Q 5 K Therefore, the accuracy of the calculation may be checked by verifying that the symmetry relation (11.10) is obeyed. II.B.3. Angular Correlation Function and Spin Flip The angular correlation of the de-excitation gamma radiation with respect to the direction of the scattered particle has the form (Sa 60,Br 62,Go 62): 412: Q W(9¢,9r¢')= 2.? 21:”, (1Q E, Y“ (9,39,) (11.12) where YE is the usual spherical harmonic. The parameters PK can be written in terms of tabulated (Bi 53) correlation coefficients: F = 2 CLCL' F“ (LL’szf) (11.13) K. LL' Here, Jc is the nuclear spin after the emi851on of the . - ' the gamma ray and L,L' are its multipole orders. CL 18 "T‘ In “4 ‘1- 13 probability amplitude for 2L - pole emission and the normalization is: l 2 'CL’ - 1 (11.111) L O O The spin flip subroutine calculates W(9 ,0,90,90.) for + + . . . the case of a 2 — 0 tran51tion. Only one multipole order K 18 considerably simplified. The spin-flip probability is (1:2) can contribute, so that the computation of F directly proportional to this correlation. The constant of proportionality is derived in Appendix B. II.C. Nuclear Reaction Models The model dependence of the DWBA transition amplitude is contained in the matrix element of the interaction potential V taken between the wave function for the internal states Of the colliding pair. If V is static (i.e., if it is local and does not contain gradient operators), it may be expanded in a multipole series of the form (Sa 66): V". i ”i )= 2(4).- ”V (r 1.311.»... (94‘ a) (11.15) ass A "V" Where fig represents the internal coordinates of the target, and ; those of the projectile. The 'spin—angle' tensor Q. T is given by: 5 (to T (9% $4) = Z,<9~5"“""l°/‘)Y‘") su-m (11.16) .lstL where SS is a tensor of rank 3 in the projectile spin space (for inelastic proton scattering, s=0 or 1). Taking the matrix element of V, we get (Sa 66): "'5'. .Cr) 2’, (99‘) <5.M‘I‘M¢]Vlstm‘-J2Mi)= Z a“ .25.); A s--M xL—J ‘ ‘ <5:‘£"i.‘"clsa"i'mf> (11.17) x (3;,5 M£,M4‘M£ ”1: Mr) (£5 ",".""HM3'”£> been factored out using the Wigner—Eckart theorem. The ls reduced matrix elements of the interaction: ’25.“ . (I’ll/J}, 5:5(r>= 5:7 <5.°"5,"5.> (11.18) radial function F j(r), or 'form factor', contains the The form factor is to be computed in the context of a Particular reaction model. In the present section, we discuss three models which have been used to interpret the experimental data. The first of these is a collective mOdel in which the nuclear wave functions are taken to be the POtational eigenstates of the total angular momentum of the nucleus, and the interaction is generated from a deformed optical model potential. The other tWO are micI’Oscopicmodels, in which the nuclear wave functions are Shellmodel states and the effective interaction potential u/d m, U I». t. n.\~ 15 is the sum of two—body forces. optical potential; in this case, only the spin—independent (3:0) form factor is nonvanishing. It is assumed that the I nuclear surface be defined in body-fixed axes by r = R(9,¢) and expand in spherical harmonics: n: V: V("-R(9:¢’)= V6“ Rofl+£zm¢,mV¢(9$')J) (11.19) The multipole order of the deformation is determined by the quantum number 1. For an axially symmetric deformation, the only nonvanishing parameter is Q10 =3 P1 . The next step is to expand the potential in a Taylor series about R = R0: v... wee.) — SR. 3;. V(r-R«>+-~ where: (11.20) M , , SE = Re .22»: a—QMY1(6 ¢) The first term in (II.20) is the spherical optical potential Which describes the elastic scattering. In the flPSt approximation, the interaction is taken to be that part of the expansion which occurs to first order in the deformation 16 parameters: 9.1 AV(r—R ’ ’ V- = -R £2: angL (9 96) (11.21) m “If. 0 Y‘ We now compare with (11.15), which, in the special case s=0, takes the form: ’ 1% .. (9 V5“. =1i() Kai/A 1:, W (11.22) I I Let R be the rotation which takes body—fixed axes (9 ¢ ) into space-fixed axes ( 9 ¢ ). The transformation is (Ba 62): m i "I I I e (E (9 )2 (94>) 5’ Dr)”, )1; ‘7] (11.23) where D is the usual rotation matrix. Substituting into (11.22): Comparing with (11.21) above: I’M 1 AV("'R ) 2(__) V D :: TEO TIT—J a (11.25) M 101/“ ”fr“ 1"! . . . 1 1* Uglng the orthogonality relation 5 D 1'”, :2 SM“. A4 V = '- 3“! id 31*” AWnRjFJJ (11.25) 101/.“ m In up,“ 17 * m-M’ .1. Finally, we have the symmetry relation Dnm' ‘6') I?“ -M and, since 1 is integral: J .£+M 4V7r-Eg) \{eu’u t Effie)a‘1mDM4¢-]ET-(.11 27) The form factor is proportional to the reduced matrix element of (11.27) taken between the initial and final nuclear states. Here, we shall only consider axially symmetric states, for which the wave function is (Ba 62,Da 58): 3' . __ 21'...” (330,1)? ) Ll): ,, 1): I Do I (11.28) M The matrix element of V between states with spin Ji 101,/L and Jf is: L )1” JVlr-R) ~—. (sf Mf I Ylog’uljc'Mi> "‘7 975‘ [54 Re T 'W‘E-HMHp') (00;: 212:.- ‘m (11.29) I I _ =(_)I"" {*F‘ R0 dy<flqootao> \o A ‘1) V r. (11.35) t "’ - (qr ( I/fi(...-r.1)-£g\g, no)!" Substituting this result into (11.15) and applying the the two— ' , definition (11.16) of the spin-angle tensor T153 body potential becomes: J-vu . ~—- . w I .‘P’ 2 (-) )4: (for?) 76:14 151;,“ (11.35) I 4/? : ‘ 151/“- The form factor is then obtained from (11.17) and (11.18): F m». . .12 <32“?‘£.“’fr>1..“"’i> mm (55 In order to calculate the form factor from (11.37), it is necessary to determine the initial and final state nuclear wave functions, and the interaction potentials VlS(ri,rp), from some nuclear models. The interaction potentials will be discussed in this section, and the discussion of the nuclear wave functions will be deferred to the next section. Two different interaction potentials were used in the calculation of the form factor. The first of these was a Yukawa interaction: V,“fi“rr’)= v; ('9’ S /MSR) (£=‘¥'$‘)(11.38) Where the strength V8 and the inverse range mS were In. v: 3.. T. i!!! 4V. 21 determined in the impulse approximation by McManus and Petrovich (Mc 67,Pe 67). They fitted the Fourier transform of a single Yukawa to the nucleon—nucleon scattering amplitude calculated from the central part of the Hamada—Johnson potential (Ha 62). The interaction so determined is complex and spin—dependent, and both the range and strength parameters vary with the incident proton energy. The particular virtue of the Yukawa—type interaction is that a closed form for the multipole expansion exists (Me 65): * ) 19> '(‘ r if" (r‘) ( _-: O J (M M r ' .3333 C /m R 8' 1M . + . . where 31 and hl are the spherical Bessel and Hankel functions, r>(r<) refers to the greater (lesser) of ri and r , and R 411:? I. Then: 1 P 145 (0,19) : Vii Vs éumfl’UJi? (£1113) (11.n0) The second form for the interaction potential was derived by McManus and Petrovich (Mc 69) from the Kallio- Kolltveit shell—model effective interaction (Ka 6H). The resulting interaction was real, spin-dependent, and independent of energy. In addition, it also included a factor depending on the two—thirds power of the nuclear matter density, which seems to improve the agreement between theory and experiment (Gr 67,La 67). The radial dependence is an exponential form, so that the multipole eXpansion is 22 somewhat more complicated. For an arbitrary function of R =l53—Egl: V J- V (r- r P(mq’) 3(12) = 7,, 21211)) Is :11?) 1 (11.111) 1 where o( is the angle between ri and rp Using the properties of the Legendre polynomials (Me 65): S§’(€nd)\4(R)&(md) -.-_. 9-1;}, (20.7.0411) ZSC51’rP)jgmuj g,(m.1)1(m¢) (II.H2) .!.. (r. ‘-=' .277 V15 HT) 1, 1 Next, make a change of variable, noting that R: (H.317, "JC-Co‘oad) 1: n+r 277 ‘ P ‘9 V ((2)9): nr I P302) 16(2) R R (11.113) 15 l P 112-17»! Thus, the evaluation of the multipole coefficients of the potential introduces an additional integration which must be performed numerically. A computer code has been written (Fe 68) to calculate the form factor in this case. II.C.3. Nuclear Wave Functions The excited states of doubly-closed-shell nuclei C may be expressed as coherent superpositions such as Of particle—hole states obtained by promoting a nucleon from a closed shell jh to an empty shell jp. We define the particle—hole state by (Sa 66): I()"'j).m> e 2 <3 1'. M-MMIIM) 1. P 1. (11.1111) ”1 . J 4-M‘m ‘l' 41 0.. XL“) QJ ,m—M I’m IO> .4 An V: .._~. 23 where [0) is the particle~hole vacuum wave function and + ) creates (destroys) a particle in the shell model a. (a. 1m 1m orbital j,m The excited state 'JM>’is then: C ("" J'M UM): Z I 11‘5”) (11.115) J J.) 4p 4P is the amplitude for the hjp where the coefficient Cj corresponding particle—hole pair in the excited state wave function. The amplitudes we used were those computed in the random phase approximation (RPA) by Gillet and Vinh Mau (Gi 6H), which include the effects of ground state correlations The form factor is proportional to the reduced matrix element: M -’-' < If” ‘Z If, LJJU/I 0) (11.115) In second quantization notation: . + .(L)= V" 7-. 'rn (A 0L1 eggs"; Z <37 fill/(s (sJ’J,( a) in gm (H.117) «(p 11 F “’ where the summation extends over all shell—model states Taking the matrix element of this operator between the states (11 H5), and using the commutation prOperties of the creation—annihilation Operators (DeB 6H), one obtains A ,-I Z 31‘.) Cy-in'm (11.118) P M A . Where j = fij+l. Expressions for the Single—particle 211 matrix elements .- 5%: (xi-‘0?) (gag) I} 41;. (11.119) where uj is the radial part of the shell model bound state f wave function, and j stands for the quantum numbers (N,l,j) where N is the principle quantum number, 1 the orbital i angular momentum, and j the total angular momentum of the m shell model orbital. We have used harmonic oscillator bound state wave functions, which seem to give an adequate representation of the shell—model states (Mc 69a) and have the advantage of being analytic. The nucleus l2OSn has been studied in the quasi- particle model by Yoshida (Yo 62). Since a quasi-particle is a mixture of a particle and a hole state, the expression for the matrix element M in this.model is very similar to the preceding results for particle-hole excitations. In the notation of Yoshida: A 9-! _ . . , f . J . 1. Z, +1 " 12.1w 1 (.10 112.11.11.11 J[(11.50) J Where expressions for the normalization coefficients 2:4, and ¢.J. are given in Ref. (Yo 62) and Uj and Vj are the JJ ‘ . 1 O O . usual occupation parameters (Ba 63). A derivation of this result is given in Appendix C. CHAPTER III EXPERIMENTAL METHOD III.A. Beam Line III.A.l. Cyclotron and Beam Analysis System All experimental data were taken using proton beams from the Michigan State University sector—focused cyclotron. The design and operating characteristics of this machine have been described in detail elsewhere (Bl 66). It is capable of producing high quality beams of several different projectiles over a wide energy range. For protons, this range extends from 20 to 50 MeV, although lower energy beams have been produced by accelerating molecular hydrogen (Pa 69). A schematic diagram of the beam transport and energy analysis system appears in Figure 111—1. The extracted proton beam was focused on the object slit 81 by a set of quadrupole doublets. Protons passing through this slit and the divergence limiting slit S2 are bent through 900 by the energy analyzing magnets M3 and MH, and then strike the image slit S3. The properties of this beam transport system have 25 _ NALoymno . AMA COQPOHnu‘m % T . . . 1., .c #wm9o>wcp opwpm cmmHQOHz one mo seam HmpaOEHnomxm .HIHHH mhswwh it .. \.. \lek :26 27 been investigated previously. In particular, the energy resolution of the transmitted beam as a function of slit widths (Ma 67), and the energy of the analyzed beam as a function of magnet field strength (Sn 66) have been calculated. Typical slit Openings used in this experiment were 0.25 cm for the object and image slits, and 0.30 cm for the divergence limiting slit. This corresponds to an energy resolution of 8 parts in 101‘1 full width at half maximum (FWHM). The energy of the transmitted beam was calculated to within :0.l MeV from the measured field strengths. III.A.2. Beam Alignment The analyzed beam was deflected into the appropriate experimental area by magnet M5, and was focused on the target by the final quadrupole doublet QS. No collimating slits were used near the target in order to keep background radiation in the experimental area to a minimum. Instead, the beam was positioned by observing a 0.125 mm thick piece of Pilot—B plastic scintillator* at the target position using a closed circuit television system. In practice, the excitation of magnet M5 was set to the calculated value appropriate to the particle and energy required (Sn 66). Fine adjustments were then made to center the beam spot on fiducial marks inscribed in the plastic scintillator. In this way, the beam could be centered to within 1 mm. The scintillator was inserted into the beam L fi— *Pilot Chemical, Watertown, Mass. 28 line several times during the course of a run to check against centering drifts, which did not occur. During the later runs with the tin target, the fields at magnets M3 — M5 were monitored with NMR fluxmeters. Beam spots were typically rectangular, with a height of 2 mm and a width of u mm. Angular divergence was less than :0.5°, as determined from the maximum possible beam diameter at the quadrupole doublet Q5. III.A.3. Targets The target used for the 12C experiment was a 26.5 mg/cm2 graphite foil*. Its uniformity was determined to be better than :1% by monitoring elastic proton scattering from various areas of the sample. The tin target was a 9.9 mg/cm2 l2OSn obtained from the foil rolled from 98.H% enriched isotopes division of Oak Ridge National Laboratory. The energy loss A5 in the graphite target was 500 keV at 26 MeV, and 350 keV at #0 MeV. The corresponding value for the tin target was 100 keV at 30 MeV. The mean proton energy Ep was determined by subtracting 4&/2,from the energy determined by the beam transport system. III.A.H. Target Chamber The targets were mounted in a small Al chamber in which provisions were made to mount two targets. The angle Speer Carbon Co.,Inc., Carbon Products Div., St. Marys, Pa., Shield Grade 9326. 29 of the target frame relative to the beam line could be remotely adjusted and read out to within 11.. Scattered protons passed through a 0.125 mm thick Mylar window in the side of the chamber. Energy-loss straggling in this window was approximately 180 keV. Because the states of interest are well separated (> 1 MeV in the case of 120Sn and >3 MeV in the case of 12C), this broadening is quite acceptable, particularly since the use of a window leads to major design simplifications in the apparatus which positions the proton detector. For example, the arm on which this detector is mounted does not have to be inside a large vacuum chamber. The counter arm could be remotely positioned to within :0.l° over the angular range from 25° to 155°. The limits of this range are determined by the geometry of the target chamber and beam line, and by the size of the detector package. The center of rotation of this arm coincides with the center of the target frame to within 0.05 mm, and the maximum backlash in the positioning apparatus has been measured to be i0.1° when the drive mechanism is prOperly adjusted. III.A.S. Charge Integration Protons passing through the target were collected in a 7.5 cm diameter by 1.5 m long Faraday cup located so that the beam stOp was 2 m beyond the target position. This distance was chosen to enable the elimination of chance coincidences with background radiation coming from the w ~-- ~v—w— «— 30 Faraday cup (see Sec. III.D.H). The length of the cup ensures that protons which suffer multiple collisions in the target will still be collected, and it also reduces the probability that electrons produced in the cup will leave it. A study of the background radiation produced by various beam stop materials indicated that graphite was best suited for this purpose. The observed reduction in background (about a factor of 2.5 compared to Al, for 30-MeV protons) overweighed considerations of the additional radiation hazard posed by the production of relatively long—lived (4’20 min.) activity in the beam stop. However, Al was used for the beam stop in the 12C runs to avoid possible confusion of u.uu MeV gamma rays from the target and the beam stop. The beam current and integrated charge were measured using an Elcor Model A310B current integrator. The accuracy . . . . + of this instrument has been measured to be Within —1% (Ko 67). III.B. Detectors III.B.l. Gamma—ray Detector Assembly The gamma—ray detector used throughout this experiment was a 5 cm diameter by 7.5 cm long NaI (T1) scintillator mounted on a RCA 8575 photomultiplierfi. The energy resolution Of this detector has been measured to be 7.6% for the 662 keV 137 .gamma line from Cs. ___-fir___ Obtained from the Harshaw Chemical Co.; Cleveland, Ohio. ~.>v 31 The prOper distribution of bias voltage to the dynodes of the photomultiplier was maintained by an ORTEC 265 phototube base. A schematic diagram for this unit appears in Figure III—2. One important feature of the design is the fact that the anode is operated at ground potential. This means that the fast output can be direct—coupled and therefore that the rise time of this signal is not limited by the time constant of a coupling capacitor. A minor disadvantage of i this arrangement is that the photocathode is operated at a high negative potential with respect to ground so that the outer glass envelope of the phototube must be well insulated. A major problem associated with the photomultipliers used in this experiment was the gain shifts observed at high count rates. The additional current drawn through the voltage divider string during a pulse causes an increase in the dynode potentials with respect to the photocathode, resulting in a net increase in the effective gain of the system. Since the current amplification of photomultiplier tubes is proportional to a very high power of the interdynode potential (Ch 61), this effect can place rather severe limits on the allowable variations in count rate. The problem can be eliminated for short term fluctuations, such as those due to the pulsed nature of the cyclotron beam, simply by connecting rather large capacitors across the latter stages of the tube to serve as 'charge reservoirs'. This method is effective in eliminating fluctuations with time constants of several milliseconds. It becomes quite ‘32 . .oommmccmh .mmcwm xmo m.UCH . . . i ii. mi] h. g. .053 V8-3; 3.3 .86. BQN 4308. J‘A‘XIIUJ .UMHmo mo cowmmwficmm he pmycwsmv mmmc.hoflfimwpddsowocm ._ I wow ch02 UMBmo one we Emcmmwp pwsoaflo .NIHHH mhsmflm .Ouk “KS. akgx 0.“. vb .3 Htfikwk- 0 .V B l . 4.... «5.33;. a 9.6 039... «33.386» 63% §x~§u 3.2. .... N .383 83g. fins“. V , m nuns“ «IQ -~\.H~E\ m r to . unrucuwennx. q a u w A r q... ‘— o x: _ v. y . . . I r L 5.! 33.5.! .. 3% it A la. sue JV 3 “We 13! . r 2 MWIW awe if a u 15 1 fl h a? k3 is.» xxx. 41 4 a R A 3 B: 1.4.. x! 6%.. PS: .39. Ht . HO 3. W91. 3. t. a. l \k QM.» 1.. h dkro In no lid“ 1 J1 I- 3}. .. are. i\ .3. >2 3: 5: .2. l x. 3. am" to! to! we. re! 20.. run x3 x3 x3 t9. x2! .88 is H \ u 9.. a... an m. A. or... «m w. «I». H k* \SR- 3.3. s3 0 3‘ Q . iv ‘8! Ill .38 a. H \n/ \rf \isF \mV n \Sf w o ta a“ 86‘ 3 w 5: w 9 cl. 8. 3.33 .8 but Q. U- . I 33 impractical, however, if variations in average beam intensity can occur over a time interval of minutes, as was often true in this experiment. We have substantially reduced the observed gain shifts in the latter case by employing zener diode voltage stabilization on the last four dynodes (Figure III—3). The increased stability comes at the expense of flexibility in the selection of photomultiplier gain, since the stabilizer must be designed for a particular dynode voltage distribution. The operating voltage for the gamma detector was —2400V. The gamma-ray detector was contained in a 130 kg. cylindrical Pb shield supported below the target chamber. This shield was quite effective in reducing the count rate in the detector due to general room background. The entire assembly was centered on the normal to the scattering plane to within 4 mm and the distance from the_beam line to the center of the detector was variable between 12.5 and HS cm. Scattered protons were kept out of the gamma-ray detector by the 1.27 cm thick Al floor of the target chamber. III.B.2. Proton Detector for the 12C Experiment The proton detector for the 12C experiment was a 3.8 cm diameter by 1.9 cm thick NaI (Tl) scintillator mounted on an RCA 8575 photomultiplier tube. This unit was Packaged at the cyclotron laboratory. A diagram of the completed assembly appears in Figure III-H. Because of the hygroscopic properties of NaI, the packaging was carried Ill'li .meamflpHUEOpoca one“ mo mmp0c>c 930m vmmH was how scenaswop omppao> opowpnsmCmN m mo Emsmmem .mIHHH msswwm V0 VN L 3 m0 mN 5085100 co- _ o moxax .mmz_.mmmz_ _$z_.mmmz_ «N mote. .8238; 52.052. 3 much. mmmzzvmmz. 8mz_.msmz_ NN L) ... mmmz_+m~.mz_ mmmz_+momz_ _N 4 ‘3 BNEQN 5.82%.. E If 58% .>oom_ .21 mom does. a. 0» EN... _ 25 ”$459 3+ .wmmxoma Loyompmp GOPOQQ mo COHpomm mmOhU .:IHHH mssmflm. - ,. : ... - mar. .50.... xmmi . . 52435.8 58...”. m «2.234% I . mm..." 45.3. m . ”WW“ .So .36... fl 7%... 2.532 g J0x 0N0 .02....30 I >3... 0.0muam ... 00» 393309 e c _ WBNNVHO/SLNHOO no had an active area of 80 mm2. To eliminate multiple scattering of protons out of the sensitive volume, a 0.63 cm diameter collimator was used. The detector was operated in vacuum at 1500 V bias, and arrangements were made to cool it tx>-J7Ofib'with dry ice and methyl alcohol. Under these conditions, a typical value for the measured energy resolution was 170 keV (Figure III—6) which was quite adequate to separate the states of interest. During the course of the experiment, it was found that the cross—section to the first excited state of 120Sn in the angular region from 100’ to 155‘ was too small to enable the collection of an adequate number of coincidence events in a reasonable time. Physical limitations imposed by the construction of the target chamber and the detector package restricted the minimum target-to-detector distance to 10 cm, and the beam current was limited by pileup losses in the gamma detector (see Sec. III.D:S). Therefore, the Nal detector was used in this angular range, with a 1.26 cm diameter collimator. III.C. Electronics A block diagram of the electronics for this experiment appears in Figure III—7. Essentially, it consists of a fast-slow coincidence circuit in which the fast unit is a timeetonamplitude converter (TAG). Ml ..Pcmfiflhoaxm cm Omh oma sow SDonmmm cowosm HMUHQ>H - mmmEDZ JMZZ8. 0: .17 Enmhoumm yogi. ‘0 “-1.00 c o I to o 00 on on ‘ o o oo¥~ooo 0 ‘03" 0" CF 0 ~ J mOPUMPwo 21:5 NEEOm x EEm .mefio 208E .fitmdm.auu.uvcmWN_ _ IIJ O. O. O. co. 'BNNVHO/SlNflOO H2 .wwdopwo mocopflocaoo m 3P Mo EmhmmH 00 . I a . w! - AU x am 5 HHH mhsmwm. up...» 9598 ”EB 3:80 «398 3.6.3. . wing 35 885 ”ma—(om mwsfi ,mgam —flmflu «£250 £350 1 g . .33 fl, :1 ll llll. [lllvlll [Illiligilltil Infill} [lulltlulllllJ _ N. E $24 “2‘ — l u u o I Q . “ 89:6 10311 gm 33 54... _ _ _ _ _ _ _ _ wozmQUzao Hmdu H3 III.C.l. Fast Timing‘Circuitry Presently, there are three methods of deriving time information from a detector pulse. These are: 1) leading edge trigger, 2) conventional drossover timing, and 3) 'fast' crossover timing. The last method is applicable only to scintillators havjing a decay time—constant gilOns, such as Pilot-B plastic scintillator. For this reason, it was not considered for this experiment ( t; for NaI = 250ns) and it will not be discussed here. Of the remaining methods, which are illustrated in Figure III-8, leading edge trigger is by far the better choice for fast timing since it results in less 'jitter', i.e., time dispersion due to electronic noise and statistical fluctuations in the detector pulses. Recent theoretical studies (Be 66) have shown that leading edge timing is better than the conventional crossover method by about a factor of 1H. A major problem in the use of leading edge timing is the need to restrict the dynamic range of pulse amplitudes accepted to avoid the time resolution problems associated with 'walk’ in the low-level discriminator. 'Walk' is defined by the time shift due to a fixed low-level threshold and variable energy pulses (see Figure III-8). For a pulse rise time of 10 nsec., and a ratio EthreshxlEO = 0.2, which represent typical values for this experiment, the dynamic range must be limited to D 5 2 to obtain the required 1 nsec. resolution (neglecting jitter). This is not a particularly restrictive requirement, but it does mean that vawonlpl- dazq um -.Q Fu\luCI III.D- ln“‘ “0 II I‘ll Lu; LEADING EDGE Q CONVENTIONAL a TRIGGER g CROSSOVER 25' E ' :3 ___ .1 '- _ a. g 3 DJ 3 15355390 ... —- ..l —_ — O. | a L t. (11MB SIGNAL) °‘ ILatrium: SIGNAL) TIME (1) TIME (I) Q JITTER DJ CD § § 3 __5I1I3.E§.H g b-JITTER TIME (I) TIME (II E0+AE DYNAMIC RANGE (D) = E0 E(I) = EW‘x I/r (r= PULSE RISE TIME) WALK= tTHRESHIEOI- ITHRESH (Eo+AE) - ,E [.-l] ' * E0 D Figure III-8. Schematic representation of leading-edge and. conventional crossover timing, and the definition of 'walk' and ‘jitter'. I}. , 7*!4‘ U) (“D ’r‘l 2 v. Q .- HS the fast coincidence measurement must be supplemented by side—channel amplitude analysis. The crossover technique does not suffer from this problem since the crossover time of a double—delay—line clipped pulse is independent of the pulse amplitude, to a very good approximation. Therefore, it is particularly valuable for those applications in which a very large dynamic range is required. In this experiment, timing information in the fast coincidence channel was obtained by leading—edge timing with suitable low—level discriminators viewing the fast signals from the detectors. The logic signals from these discriminators were used to start and stop the TAC. A variable delay was introduced into the stop side to shift the 'true + chance' peak in the time spectrum to a convenient pulse height. The time spectrum was gated by a logic signal from the slow coincidence unit, thereby restricting the effective dynamic range as discussed below. The gated time spectrum was analyzed using a 102a channel pulse height analyzer. Typical time spectra are shown in Figures III—9 to III—l2. III.C.Z. Side—Channel Amplitude Analysis The slow coincidence unit, which provided pulse-height information, was a conventional zero—crossover coincidence circuit. The linear signals from the two detectors were converted to bipolar pulses and analyzed for both pulse height and time information using timing single channel may 90% u HHH madman MO Pasosao mfiw £PH3 Umfladpno Enavommm wasp HMUHQ%H M6 . m2 Eonw mxmsnmafimm >ox Ham. mm mmmEDz szzm2 «.64 was 2.3..” u m: . . I ma . 00 as monommo >msumEEmm any 90% UmcflMpno Enhvommw wasp Hmofimxfi 0H HHH on: .m mmmEDz szzH0mmm smomcsn ohm mxmmm A.mozm mocmno .QPUfiB mmadm Soon ogp Eosw COfivdnflnpcoo m OP map .Azmzmo 0mm: H.H ma xmma A.moz52 JMZZ.. ......«.. .. . .. NR... 5... ...... as ............. TV a... .... i. 52 'IBNNVHO /S.LNFIOO 0mm JwZZP m wadzocw ”PCCEHLwaxC o . NH mcp 90% Essvomam hmpI IMEEmw awn—232 J.II._ZZ2¥u2:$ 5&2 \ ..Mwwnmo 0238303 E33 ozEwEBm 29128 9 I . I 0mm 5 >22 v.3. mOhomhwo 2.5—oz ..mx..N SSEDMQw >o2 woman ... Aimvfom. pa .309 amass >02 mm... _ 73 NNVHO/SINIIOC) 00m _ I.) ~-‘ .pu 60 Table 111-1. Measured values of the efficiency of the gamma- ray detector for the 12C and 120Sn experiments. Experiment E (MeV) E (MeV) Ln. (sr)a) 6 m(sr)b) P Ir r V t 12C 25.2 u.uu 1.23x10’3 (5.10:0.55)x10'3 120 10.0 u.uu 1.23x10‘3 (5.u1io.76)x10’3 lZOSn 30.0 1.17 8.72xlo‘2 (2.2IIi'I).11)x10"2 a) Measured at the center of the 2" x 3" NaI (Tl) scintillator. b) Efficiency for gamma rays in the pulse—height window. Table 111-2. Measured values for the strength of the 60Co source. EBlza) Counting Time(hrs) Source Strength(sec—l) 180° 3.5 (8.uuio.20)x105 180° 3.5 (8.3u:0.20)x105 + 90° 11.0 (8.52-0.1u)x105 Average: (8.51:0.15)x105 u ) Angle between the two detectors, one Of WhiCh was a 3" x 3" NaI (Tl) scintillator at 18" from the source (measured to the center of the detector). The other detector was a 2" x 3" NaI (Tl) scintillator at 12" from the source. . -.-. ...- .- L/W F . 5 v voo v- .p‘ Lu ‘b ‘0; ‘1 61 For the l.l7 MeV gamma line from 120Sn, the well—known method of ‘7““7 coincidences following the decay of 60Co was used to prepare a standard source. The relevant decay scheme appears in Figure 111—17. The 5.26 year ground state of 60Co decays by F“ emission. greater than 99% of the time to the 2.50 MeV 9+ state in 60Ni. This state subsequently decays via a gamma cascade through the 1.33 MeV 2+ level. The two gamma rays, with energy 1.1? and 1.33 MeV, are always emitted in prompt coincidence. The angular correlation function for this cascade is well known (Kl 53,La 53). Suppose that we now prepare a source of strength S (unknown) and then observe coincidences between two_gamma detectors which count the 1.17 and 1.33 MeV gamma rays, respectively. The number of singles events in the two detectors is: St fig = (31 A111 «$5; (III.6) 3: N1 7" 61 Ana. ’7, in a time t. During the same interval the number of coincidences is: §3. 49 ) e bqu Nc’ E'A‘Q' 97" [W(‘ T 2' (111.7) 63 where W( 912) is the angular correlation function and 12 is the angle between the detectors. These equations may be solved for the source strength in terms of known quantities: ‘ITl' W(9n) E El: ' t A, (III.8) 62 _z .000 90% mEmSUm hmomQ .hHIHHH mfismflm . . 00 m _ mmmfl. ilj 633.9 N $90.0 .xzmm 3»:qu ... .28 8.9 .. 008 m 63 The method is potentially quite accurate. The results obtained for two values of 9 l are listed in Table III-2. 2 The calibrated source was then used to determine the 120 efficiency of the gamma detector for the Sn experiment; the measured value appears in Table III—1. III.D.3. Acceptance—Angle Corrections In any real correlation experiment, the effect of the finite acceptance angles of the detectors on the measured correlation function must be considered. In the case of the present experiment, a gamma—ray detector subtending a finite angular range will accept some radiation from the A m=0 and ‘AUFi2 transitions (non—spin-flip events). If the proton detector also has finite size, the normal to the scattering plane become somewhat indeterminate since this detector then defines only a range of scattering planes. Corrections can be made for these effects if the complete correlation function W( g,9’., ¢T) is known (Sc 6”). However, the experimental determination of W is difficult since it requires coincidence measurements as a function of three angular variables. Fortunately, it is possible to deduce minimum and maximum values for the finite-aperture corrections from the gamma—perpendicular correlation alone (see Appendix B). The actual correction is taken to be the average Of the maximum and minimum values, with an uncertainty equal to i3u% of the difference between them. That is, in the absence of better information it is assumed that the 6; 'true' correction will be uniformly distributed between the maximum and minimum value. For angular acceptances which are 'small enough', the correction (and its uncertainty) is also small. The interpretation of 'small enough' depends on the degree of difficulty of the experiment. It is always possible to reduce the angular acceptance to the point at which the correction becomes negligible. However, this involves a decrease in the coincidence count rate which cannot in _general be regained by increasing the beam current because of the associated increase in the accidental coincidence rate (see Sec.III.D.U). Therefore, an appropriate balance must be struck between the uncertainty in the acceptance- angle correction and the statistical uncertainty in the number of coincidence events obtained in a given time. In the case of the 12 C experiment, it was possible to reduce the angular acceptance of both detectors to :30 and still maintain a reasonable coincidence count rate. The uncertainty in the acceptance—angle correction was therefore quite small. For example, the correction for a spin—flip probability 81:0.10 was (—U.8:2.3) x 10_3. For the more difficult 1208n experiment, it was necessary to increase the acceptance angle of the gamma detector to :10° to obtain a reasonable count rate. The proton detector subtended a :20 angular range because of physical limitations imposed by the small size of the Si(Li) detector. The corresponding correction for 81:0.10 was —(2.1:0.8) x,10—2. 65 All experimental data were corrected for the effects of finite angular acceptance, and the associated uncertainty was included in the quoted experimental uncertainty. III.D.H. Accidental Coincidences The accidental coincidence rate between two detectors viewing a continuously radiating source is given by (Gr 66): (MA): (”1} 2’ (111.9) where (N1) and (N2) are the average singles count rates in the two detectors and ‘Z’is the resolving time of the coincidence circuit. The same formula is applicable to the case of a pulsed source such as a cyclotron beam, except that must be replaced by an 'effective resolving time' 15‘”. which is generally much larger than t’ . For example, it can easily be shown (Hr 67) that for a coincidence circuit with resolving time 25 which is less than the time T between beam bursts and greater than the beam pulse width b: eff- (111.10) Typical values for these parameters in the present experiment are ’C’ = 1 nsec, b = 1 nsec, and T = 60 nsec. Note that be“, is independent of 2' if bétf-T so that 50 nsec is 'as good as' l nsec. Of course, this is only true if there is no 66 possibility of chance coincidences occurring between beam bursts. Figure III-ll illustrates the fact that this is not generally the case. There is a finite probability for‘ chance coincidences to occur while the beam is off due to stray background radiation between beam bursts. Of particular interest is the small peak occurring midway between the major peaks in the time spectrum, which is due to a component of background radiation coming from the Faraday cup. By adjusting the distance from the target to the beam stop so that these events arrive between beam bursts (see Sec.III.A.5), it is possible to eliminate them from the chance coincidence rate (along with a major portion of the remaining continuous background between bursts) ,merely by placing a sufficiently small window around the 'true coincidence' peak. However, it is also clear from Figure III-ll that the reduction in the chance rate obtained in this manner is small compared to that which could be obtained if the resolving time were made smaller than the beam—pulse width. The preceding discussion of the calculation of chance coincidence rates is valid if the beam pulses are perfectly uniform over the counting period. This is in fact not always the case with our cyclotron beam. Modulations with frequencies of 360 Hz and 100 kHz have been observed. Corrections can be made for the effects of these_modulations. The result is invariably an increase in the effective resolving time (Hr 67). The formula for 3"“... in the _general case is quite complex, and it includes terms 67 depending on the specific nature of the modulation which are difficult to measure accurately. For this reason, the following method was used to obtain an accurate measure of the number of accidental coincidences in each experimental run. The elastic and first excited state inelastic events were counted in singles and coincidence. Since the elastically scattered protons cannot be in true coincidence with a gamma ray, they provided an accurate measure of the accidental rate. If NOs(N0c) and le(N1c) are the total number of singles (coincidence) events for the elastic and first excited state inelastic scattering, respectively, we have for the accidental coincidences: N = N (—-—) (111.11) since the probability that an inelastically scattered proton will produce an accidental coincidence is exactly the same as that for an elastic event. Although the method just discussed enables accurate chance coincidence subtraction, it is still desirable to reduce the chance rate to a minimum to obtain increased statistical accuracy in the data. This was particularly true for the 120Sn experiment, in which a low true coincidence rate and a high accidental rate due to the large background Of gamma radiation coming from the target conspired to make data collection more difficult. For this experiment, a significant reduction in the accidental rate was achieved 68 by monitoring the gamma detector output on an oscilloscope set to 50 milliseconds full scale horizontal deflection and triggered at the line frequency. Under these conditions, any 360 Hz.modulation of the beam was clearly evident from the time distribution of detector pulses. Minor adjustments to the tune of the cyclotron and to the external beam handling system were then made during the course of a run to keep the modulation to a minimum. In practice it was possible to eliminate the 360 Hz modulation almost entirely by careful adjustment. Some idea of the improvement in the accidental rate obtained in this manner may be inferred from a comparison of the observed number of accidental coincidences to that calculated from (111.9) using an effective resolving time equal to the period of the beam bursts. The average value of the ratio of these quantities was 3.88 for the unmonitored runs, and 1.36 for the monitored runs. Finally, it can be seen from (111.9) that the accidental rate is proportional to the product of the Singles rates in the two detectors and therefore to the square of the beam current. This fact may be used to determine the optimum beam current to use in order to obtain the best statistics on the number of real coincidences in a given counting time. Let T be the total coincidence counting rate, and let A be the accidental rate. Then R = T—A is the real coincidence rate. The fractional error in the total number of real coincidences obtained 69 in a time t is determined by the propagation of errors: Err-(EU l -" J- 2A. __.._.._.,. E?- J(T+A)t = t ‘ J:* 75" (111.12) 2t Since R is directly proportional to the beam current I, while A is proportional to 12: errUEt) -'r,_ [4, j Rt t "f + Jel- (111.13) This rather surprising result indicated that, everything else being equal, the best statistics on the number of real coincidences is obtained for very large beam currents such that the first term in (111.13) vanishes. Actually, other problems such as dead—time losses at high count rates and the effects of beam modulation have to be considered so that it is not advisable to use extremely high currents. III.D.S. Pulse Pileup and Dead Time Losses There are several ways in which coincidence events can be lost due to dead time or pulse pileup in the coincidence circuitry. For example, the TAC has an average dead time of u/flsec every time it is started. Losses from this source were reduced by starting the TAC with pulses from the detector having the lower average count rate. Even so, the average start rate in some cases was as large as 10‘4 sec—l, which corresponds to a dead time loss of 4% (for an unmodulated cyclotron beam). This was the 70 major source of dead time losses for the 12C experiment. Coincidence losses may also occur if a time conversion started by a 'true' event is stopped by an accidental coincidence from a preceding beam burst, or from the uncorrelated background between bursts. The probability that this will occur depends on the location of the 'true coincidence' peak in the time spectrum. For the 12C experiment, the maximum equivalent dead time introduced by this effect was 200 nsec, which corresponds to a . . . . u —l negligible 0.2% counting loss at the typical 10 sec . 120 count rate. Because of the large background in the Sn gamma—ray spectrum (see Figure III—18), the average stop -1 rate was higher in this case (”’105 sec ). Therefore, the TAC conversion gain was set to 100 nsec full scale so that only events from the 'true coincidence' peak and a part of the continuous background between peaks were analyzed. The equivalent dead time from this source could then be neglected. It should be mentioned that the recovery time of the fast discriminators was very small (AIlO nsec), so that c01nc1dence losses due to the dead time of these units were also negligible. Finally, coincidence losses can occur due to pulse Pileup in the slow coincidence circuit. The shaping amplifiers used in this experiment had time constants of ° 1 0.25/u.sec, so that the total Width of the output pu se was approximately 1 [lsec. Two pulses arriving within this interval are algebraically added by the amplifier . HMflvcocomxm mzv wCHBO£m anoEwoomxo cm 71 .Am.Q.HHH.ome mcofiwmasoamo asmawm may CH poESmmm COfipSAflopmflp OmN omH $39 90% Esopowmm >MQIMEEMQ ”Ham—>52 IEZZ22 5 \ l 1.0. >02_O; ankommm >220mu m £5.58. a. e .58. _ _ mo. '"IBNNVHO/SanOO an -r »:I “Y .1- U- . I 72 circuitry. The resulting output pulse will be rejected by the coincidence unit if it fails to satisfy the pulse height requirements or if the zero—crossover time is shifted by more than :0.5/usec (i.e., half the resolving time). This property has been used as the basis of pileup rejection circuits. A FORTRAN—1V code has been written for the SDS £7 computer at the cyclotron laboratory which simulates the behavior of the coincidence circuitry to calculate the rejection probability as a function of the count rate and the width of the pulse—height window. Two approximations were made in this calculation. First of all, the pulse shape at the output of the shaping amplifier was assumed to be of the form: Act): A. *4, (oststa) 0 A0 (2 —- Wt.) mst $2.51.) ..Lstathq] A(t) ...fle, (e ) (2.56,,sts/ota) = 1 Mt) (111.1u) where A(t) is the amplitude of the pulse at a time t and t0 is the amplifier time-constant (typically 0.25 /lsec). This is a reasonably good approximation (see, for example, * . the ORTEC catalog #1001 ). Secondly, the amplitude spectrum Of the detector pulses was described by an exponential distribution (Figure III-l8). A set of pulses with this amplitude spectrum was distributed in time according to the interval distribution (Ev 55) for the average count rate. *ORTEC, Inc.; Oak Ridge, Tennessee. .- q; v\ gn t; (I) 73 Whenever one of these pulses overlapped a reference pulse corresponding to a 'true' event, the pulse shapes were added algebraically and the resulting sum pulse was rejected if it did not meet either the pulse—height or the zero—crossover time requirements. The results were used to correct for coincidence losses due to pulse pileup. A typical value of the rejection probability for the 120Sn experiment was 2% at an average count rate of 105 sec-1 and for an unmodulated beam. In the previous discussion, an unmodulated cyclotron beam was assumed. In point of fact, the beam was not steady (see Sec.III.D.U). A correction must be made for the effects of beam modulation. Fortunately, this correction can be expressed in terms of the ratio of experimental to calculated accidental coincidence rate. Let 2; be the dead time of the coincidence unit, and let a(T) be the average count rate over the interval between T and T+dT, where Cfl7>°'tg. It is also assumed that dT can be selected so that the count rate does not vary appreciably over this interval. The fraction of intervals shorter than. 2; (i.e., the fraction of lost events) is (Ev 55): --O.('r)‘l‘:d RT) 2 ,_ e x OAT) "ta (111.15) The approximation is quite good since a(T) Ta is typically < 0L1. The total number of events during dT is: NT = a(T)dT (111.16) 74 and the total number of lost events is simply: NL = a2(1')73;d'1 (111.17) The fractional dead time loss f is obtained by integrating NT and NL over the counting period and dividing: 7% t AT 'L 1‘: ECU” ’C’d : <——-——-°' >1 (GO ’6; €90.07“. ((1,) (111.18) 0 Since a(T) is proportional to q (T), the average charge per beam burst: 2 {‘=’ <":'—""?">'z.. <:CL>”t; <9.) (111.19) In the case of uniform beam, the first factor reduces to one. For a modulated beam this correction factor can be easily related to the ratio of experimental to calculated accidental rates by applying a parallel argument to (111.9) above (Sc 6h). The resulting expression for f is: N E" “ < ’t— ’t’ = Q> : 10 N n. d P° at (111.20) A Where NATh is calculated from (111.9) and (111.10). The correction factor fO was applied to all dead—time and pulse PileUp corrections. The maximum value of the corrected dead time was 1u% for the 12C experiment, and 8% for the 120 , Sn experiment. 4 75 III.D.B. Analysis of Experimental Uncertainties The major sources of error in the determination of the relative spin—flip probability are: (l) the statistical uncertainty in the number of real coincidence events, (2) the uncertainty in the solid—angle correction, (3) the uncertainty in the dead time correction, and (H) possible errors in the positioning of the two detectors. The first of these always made the largest contribution to the final experimental uncertainties quoted in this eXperiment. The expression for the statistical uncertainty in the number of real events is readily derived using propagation of errors. In the notation of Sec.III.D.H the number of real events R is; 72%... - N... (:15) as (111.21) so that the uncertainty in R is: 7‘fi Nae N1,S 7—(1 +J— +J.) w (R) ‘ Wu" ( '77:") ”ca ”05 1V1; (111.22) 0 Since the number of singles events is always large compared to the number of coincidence events, this reduces to: N A] I J. ‘ if‘ ......) ... Ju..+( 3:) (A4,) -= JE—A; The uncertainty in the solid angle correction has been discussed before and a derivation of the relevant 76 expressions appears in Appendix B. These formulae may also be used to determine the contribution from angular positioning errors. The estimated magnitude of these errors has already been mentioned in Sec.III.A and Sec.III.B. The corresponding uncertainties in the spin—flip probability were negligible in all cases, since the angular positioning uncertainties were always much smaller than the angle subtended by the detectors. The uncertainty in the dead time correction is more difficult to determine. In particular, the quantity NATh appearing in (111.20), which is the number of accidentals to be expected with an unmodulated cyclotron beam, is somewhat uncertain due to complications introduced by accidental coincidences occurring between beam bursts. These events were neglected in the calculation of NATh. It is estimated from the time spectrum presented in Fig. III-ll that a possible :10% uncertainty may be present in the calculated value. The corresponding 10% uncertainty in the dead time correction made only a very small contribution to the quoted errors, since the correction itself was always small (see Sec.III.D.5). The only major source of error in the determination of the absolute spin-flip probability was the uncertainty in the gamma—ray detector efficiency. Contributions from other sources such as the target thickness or the current integration are canceled in taking the ratio of real Coincidences to singles. In the case of the 12C experiment, the error in the efficiency measurement was large (10—15%) ~ 11v u 77 since it is not easy to make a calibrated source of 4.94 MeV gamma rays. The indirect method used (see Sec.III.D.2) provides many opportunities for error to creep in. The major sources of error were the integration of the proton and gamma—ray angular distributions, and the determination of the number of u.uu MeV gamma rays in the window (a significant background was present (Figure 111—16) and had to be subtracted). The uncertainties introduced were quite large as reflected in the final computed errors. The 120Sn normalization was much less uncertain because we could more easily prepare a calibrated source (see Sec.III.D.2). The only major sources of error were the statistical uncertainty in the number of true coincidences and the uncertainty in the solid—angle corrections. Both these errors were made small by reducing the solid angle and counting for a long time (”V12 hours). The resulting uncertainty in the source strength was 12%. An additional uncertainty was introduced into the efficiency calculation due to possible errors in locating the window on the gamma—ray spectrum (Figure III—18) so that the total uncertainty in the normalization was estimated to be :5%. In summary, it should be pointed out that the error bars in the spin-flip distributions to be presented in the following chapters take into account all the relative errors mentioned above and are to be treated as standard deviations. The errors in the absolute normalization are not included and must be treated as uncertainties in the indicated absolute scale. CHAPTER IV OPTICAL MODEL ANALYSIS IV.A. Elastic Scattering Wave Functions and the Optical Model The tacit assumption behind the perturbation method which forms the basis of the DWBA treatment of direct reactions is that the elastic scattering, i.e., the major part of the nucleon—nucleus interaction, can be treated exactly. In practice, this is not the case. The elastic scattering is treated in the Optical model approximation (Jo 63). The n—body problem of a free nucleon (the projectile) scattering from an ensemble of bound nucleons (the target) is approximated by a much simpler one—body problem in which the total interaction is replaced by an equivalent complex spherical potential. The real part of this optical model potential represents an average elastic interaction between the projectile and the target nucleons and the imaginary part represents the absorption of the projectile into Open channels other than elastic scattering, e.g., (p,n) reactions, inelastic scattering, etc. The general form of the potential used is: (70') = 1):") - V5091) "i (‘0’ find!) cit-91.) 430(1) +(,,-§r-,}‘(Vs.*w‘°);'f (J; 3;) Wu) 78 (IV.1) 79 The functions f(xk) are of the Woods-Saxon (or Fermi) form: " ’1 . .. A3) . (’Jkrso my“; (a “+1) 7‘;( #1.,“ ) representing a diffuse well of mean radius rkA . The 'diffuseness' parameter a is a measure of the width of the k transition region at the edge of the well where the potential is changing rapidly. The derivative of this form represents a 'surface' interaction since the derivative peaks at the mean radius. In this case, the diffuseness parameter is related to the width of the surface peak. Uc(r) is the Coulomb potential between a point charge (e) and a uniformly charged sphere of radius rCA y, and charge (Ze): L 153;“, (7“3’;A’) Uc(r)= 4,24% 1:,» (“a“) (IV.2) 2rA’3 FA, The ‘Coulomb radius' rC is taken to be 1.20F for proton scattering (Sa 67). The spin-orbit term is of the Thomas type; 1 and 0 refer to the orbital and spin angular momentum of the projectile, respectively. The normalization constant, which contains the pion mass m“., has the convenient value (2}): 1:112JO Fri u The DWBA code computes the elastic scattering wave function as a solution to the Schrodinger equation for a projectile scattering from the potential U(r). The various parameters which must be specified in the input to the program x v. awn. a .1. V! a. .fi 80 are determined by fitting the appropriate elastic scattering data. This means that the exit channel parameters should be chosen to fit the elastic scattering data at the exit channel energy (the beam energy plus the Q-value for the reaction). If taken literally, it also means that one should fit elastic scattering data from the target in its excited state. The first of these requirements is relatively easy to meet. It involves a study of the energy dependence of the parameters over a reasonably small range of incident proton energies. Since the local potential U(r) is used to approximate the nonlocal projectile-nucleus interaction, one expects a variation of the parameters with the incident beam energy (Jo 63). The effect of this variation on the exit channel parameters could be significant in the case of 12C because of the large Q—value to the first excited state (—4.HH MeV), 1208n (Q=—1.17 MeV). and it should be less important for The second requirement is clearly impossible to meet. Instead, it is assumed that the elastic scattering from the excited state is, in fact, not too dissimilar from scattering from the ground state. This neglects, for example, the effects of a possible spin—spin interaction in the exit channel where the target has nonzero spin. However, this type of coupling is expected to be small provided that the target mass is much greater than the spin of the state in question (Jo 63), and it has been shown to be negligible for nuclei as light as 2“Mg and 27A1 (R0 61). 81 1V.B. The Search Procedure Optical model analyses were performed with the search code GIBELUMP* on the SDS 2 7 computer at the cyclotron laboratory. This code varies the potential depths and geometrical parameters, singly or in any combination, to obtain a fit to the experimental elastic scattering data. The criterion imposed on the fit is the minimization of the 1 1 2 quantity 7., : 1;. t 76,, , where: 7. ”0' 7. Eli : [If 2 fLE6; “ha; v“)]/A0;rw} Ahr .r a z (IV.3) if. - t Z {[fiAUI-fivtilj/AQWW} i Nd.(Np) is the number of experimental cross section (Polarization) data points, Oz’d’and 024:“ (P1.(0 “43”,”) are the theoretical and experimental cross section (polarization) at center-of—mass angle 9" , and Agxtij (Ag/9(6)) is the experimental uncertainty in 6;”(6) (axr(£)) It is well known (Ba 69) that the Optical model parameters obtained in this way exhibit certain ambiguities. That is, there exist many sets of potentials which predict essentially the same elastic scattering. For example, if the depth of the real well V and the real radius rR are k Unpublished FORTRAN-1V computer code written by F. G. Perey and modified by R. M. Haybron at Oak Ridge National Laboratory. D! 82 varied in such a way as to keep the product Vr 2 constant, R it is possible to obtain a series of potentials which give equivalent fits (Ba 64) to the elastic data. In addition, to this 'continuous' correlation, there exists a 'discrete' ambiguity in V corresponding to the fact that potentials with different numbers of half—wave-lengths of the optical model wave functions in the interior of the nucleus give the same asymptotic phase shifts, and hence predict the same elastic scattering. Finally, it should be mentioned that these ambiguities are by no means limited to the real part of the potential. The imaginary well depth and diffuseness are closely correlated (G1 65), as are the imaginary volume (W) and surface (WD) well depths (Sa 67). The combined effect of these ambiguities is to make simultaneous searches on all the parameters unfeasible since the search procedure tends to become unstable. That is, the parameters rapidly become unreasonable while effecting no significant change in ’11:. It has been found that these 'runaway' searches can be avoided simply by doing a 'patterned' search using groups of uncorrelated or weakly correlated parameters (Pr 68). The parameters were divided into three groups, each of which contained a potential depth from one of the three parts of the potential (real, imaginary, spin—orbit), a radius parameter from another part, and a diffuseness parameter from the remaining part. The parameters in a particular group were varied simultaneously, with the remaining parameters held fixed. 83 When a minimum value for x; was followed for the other groups. The entire process was was found, the same procedure repeated until it converged, i.e., until no significant change in X% was evident after an iteration. IV.C. Optical Model Parameters for 12C(p,p)12C The 12C optical model potentials used in the DWBA calculations were determined from an analysis of published elastic cross section (Di 63,Fa 67,Bl 66a) and polarization (Bl 66a,Cr 66,Cr 66a) data taken at 26.2, no.0, and H9.5 MeV. Preliminary searches were made with volume imaginary (WD=0) and surface imaginary (W=0) potentials, and also with a mixture of the two forms. In the latter case, it was found that W and WD were strongly correlated in such a way that the search code tended to drive one or the other of them to zero, depending on initial conditions. This correlation has been previously noted for elastic scattering from 9Be and 12C (Sa 67). For this reason, pure surface imaginary potentials, which seemed to give somewhat better fit than volume types, were used throughout the final analysis. Furthermore, it was found that the optimum value for the imaginary spin—orbit depth WSO tended to be very close to zero, in agreement with previous observations (Sa 67,Gl 67, Fr 67). It was therefore set equal to zero in the remaining searches. The other nine parameters were allowed to vary, using the patterned search procedure outlined above. Since we .6 A \ l v 84 were particularly interested in spin—dependent effects, it was decided to bias the searches toward fitting the polarization data. For this reason, whenever two sets of parameters gave equivalent X$ preference was given to the set resulting in smaller X2p. The results of this bias are apparent in the X2 values for the final parameters, which appear in Table IV-l. The corresponding fits, shown in Figures IV—l to 1V-3, illustrate the fact that it is difficult to understand the 120 elastic scattering data in terms of the Optical model. It was possible to obtain good fits to either the cross section or the polarization data alone, but attempts to fit them simultaneously resulted in rather unsatisfactory compromises. Discrepancies of this nature have been noticed in previous analyses of 12C elastic data (Sa 67). It has been suggested that they may be a result of the rather strong coupling between the ground state and the first excited state of 12C. However, calculations in which the equations coupling these states were solved explicitly indicated that this was not the case (Sa 67). It seems, then, that these difficulties may be related to the failure of the assumptions of the Optical model for such a light nucleus; in particular, the averaging implicit in the potential scattering model may be an invalid procedure for a system with only twelve nucleons. The energy dependence of the parameters obtained is illustrated in Figure IV-M. The 'error bars' represent the limits over which the parameter can be varied with less 85 .mpcwom memo ACOHpmNHomaomv QoHpoom mmooo mo Loofisc one ma AQZQIQZH .Mx pcm.mx mo cowvmvsmEOo one cfi pow: ohms Asmm so UGO “mm powwow Hmnsm shame HQV cfi UmpOdv mmflwcHMPLooc: flowcoefloomxm OLE an .Ao.ouom3V Eomp pfloooIcflmm Home m was Ao.ou3v 590M hemcfimmfifl oome93m m Sees poeoowoom mews mcowwmasoamo Hmcwm one npxwp one CH pocowpcoa w< AM o mm Hmm.o mo.H mH.m mmm.o HN.H Hm.m mHs.o mo.H mm.m: m.m: ma mm mm:.o mo.a m:.n mmm.o mm.H mH.m mmw.o mo.a mo.m: 0.0: 5 mm mw:.o Ho.H :m.w mmm.o :m.H mm.m :mm.o mo.H m:.m: m.mm a z x mm. III. Om Om Lox Amv a m Ame n A>mzvom> Asses levee A>mzve3 Amvmm levee A>mzv> Asmzo m Am.m->H 0p HI>H mansmem as agonm meme Oflpwmam o ozp OP mpflm mgr poozooom gowns whopoEmomm HOUOE HMOHPQO .HI>H OHAMH NH 86 I00 I 1 I f 1 )- ‘1 . l2C+p : C ELASTIC ‘ Ep=26.2 MeV ‘ . i ‘ , 6 O {DICKENSJLOL .- § ‘ ‘ CRA|G,O'.O|. 'i be: 2 . b ‘0 IO - y. p L L 4L LO I l I t I cc.4 ‘;§é“‘ . l 9; a. 0.0 A l -l.0 L L L ' ' 0 30 60 90 l20 I50 I80 Ocmjdeg.) 12 ' Figure IV—l. C elastic cross section and polarization fits f obtained for the 26.2 MeV data (Di 63,Cr 66) with the optical . model parameters of Table IV-l. Cross sections shown in ratio ; to Rutherford scattering. 87 .L I I I l I I2C + p ' fir - 9‘ u ELASTIC I“ I _ a f. ‘0 ED: 40 MW 1’.’ ' . I BLUMBERG,et.aI. g: IO _- __ g : ' ’h J: 1 “L I I I I I 1‘ L0 l I l I I P(9) -|.0 I 1 4 ‘ ‘ 0 30 60 9O IZO |50 I80 Oc.m(deg.) Figure 1V-2. 120 elastic cross section and polarization fits Obtained for the 40.0 MeV data (Bl 66a) with the Optical model parameters of Table IV—l. 88 —- o O " O o o ‘; bu: l0 — "' '- ‘E : -. b '— -. ‘D — , . '— O O | I I I I I .0” 3r E LASTIC Ep =49.5 MeV — _ O FANNON,et.al. IIIIII I I i +0.8 '- ‘ «fCRAlG,eI.aI. i 1 Q +0.4 — I I I l l O 30 60 90 Gem (deg) Figure 1V-3. IZO ISO ISO 120 elastic cross section and polarization fits - obtained for the 49.5 MeV data (Fa 67,Cr 66a) with the optical 1 infialmnarameters of Table 1V-l. 89 .Wx cw mmcmno wmm away mmoa spas mom> coo omwofimamm mgr £0H£3 po>o mwcmh may OVOOflUCM .momn soosm. .wsOmemomm HOOOEIHMOHPQO UNA mo moampcoomp mmsocm .:I>H madman $02 am $23 a 0 $25 am 8 0.. 0m 0w. 8 0.. 0m 0m 0.. on a _ _ I 0 a _ _ owno a a omd I m o l I QC 3 I Ho l I No no a _ LII _ no CI _ _ Q0 _ _ a 0.. a _ T 0.. I ~._ _ H H I H H I 2 E l I 3 u _ _ _ .— ..Iw; _ _ . p h N.— _ a _ 0.~ _ _ _ On I 106 I 10¢ . (.../f 3.; I 10.0 I I on a 90 than 25% increase in X%. These limits were determined by searching on a single parameter, leaving all others fixed at their optimum values. A typical 'map' of X2-space obtained in this manner is shown in Figure IV-5 for the spin— orbit diffuseness parameter. IV.D. Optical Model Parameters for 120Sn(p,p)lZOSn Complete elastic scattering data (cross section and polarization) is available for 120Sn at 30 MeV (Cr 64,Ri 64). The data have been analyzed in the optical model by several authors (Gr 66a,Sa 67b). The resulting potentials are all quite similar and the fits obtained are uniformly good. A selected list of some of these potentials appears in Table IV-2. Figure IV—6 illustrates a typical fit to the 30 MeV data. The potentials of Table IV-2 were used in the 120Sn DWBA analyses. The same parameters were used in both the entrance and exit channels since there is not enough eXperimental data to determine the energy dependence of the parameters. However, the low Q-value to the first excited state (-l.l7 MeV) leads one to believe that this will not lead to serious error in the DWBA calculations. .91 IO I—‘I l l- l l l2 ' C(p.p) '2C IO— _ .02 I I I I I 0.20 0:50 0.40 050 0.60 0.70 0.80 A30 (I) Figure 1V-5. Typical 'map' of X2 space for the spin—or bit diffuseness parameter. 92 #00 003 A0 .Ao5m 000 Eosw :0x0y 0903 mp0m 90H0E090m wchH0E0h 059 509% Q0M0p 0903 mH 0cm 90GHw0EH 0gp n00000 HH0 CH mo mHHc: cH ms0H0E0o0m H00H9H0E00m 0gp 0c0 .Amo meagmu >0: CH c0>Hw 0&0 090H0E090a £HQ00IHH03 AM 0.0H 0.: 355.0 00.H mm.0 m50.0 Hm.H 0m.5 HN.N 0H5.0 0H.H C.Hm MN 0.0H m.m 135.0 0H.H 5H.w 0m0.0 Hm.H mm.5 5m.N 0N5.0 5H.H m Hm QN 0.0H mm.: 0N0.0 50.H 0m.0 300.0 0m.H 5:.5 50.N mH5.0 5H.H : Hm UN 0.mH 5.5 005.0 0H.H mH.0 m:0.0 mm.H mN.5 00.m m05.0 0H.H 0.Nm mm 0.0H m.m mN5.0 5H.H NH.0 530.0 Hm.H 0:.5 mm.m mm5.0 5H.H :.Hm H0 HeH Q3 3 m0 ME > Am .>0Z om #0 EmQ 509m mnH90HH0ow QOHOQQ OHH00H0 90M 090H0E090Q H0UOEIH00HHQO .NI>H 0Hfl08 93 LG I l I l I . _ : A I203Mp : " ELASTIC : " Ep=3cmmav .— " ' A.RnLEvAuonuumn_ ._ ‘ _ s“ o. :- A :3 :2 A? :: 1: __ _. 0.0l 1 l I I I l l I I I +118-—- § ._ § CRAKLctaL +0.4 — .. 5.: A 1 IL 0.0 V -0.4 — -0.8 - _ l l l l I O 30 60 90 l20 l50 I80 I am (deg) : Figure IV—6. 120Sn elastic cross section and polarization fits 5 obtained for the 30 MeV data (Ri 64,Cr an) with parameter set ' 2C of Table IV-2. This fit is typical of those obtained for leSn. . I‘ 0 ‘7, . A... ~ ‘1. q "A. ‘1‘, R.“ . ‘ I CHAPTER V EXPERIMENTAL RESULTS.AND COMPARISON TO THEORY * V.A. 12C(p,p')12C (4.44) ‘g r.‘£<.‘m .1. ‘lannx.’.g V.A.l. Differential Cross Sections Figure V—l shows the differential cross sections predicted by the collective model ("COLL"), the impulse approximation ("HJIA"), and the Kallio—Kolltveit interaction ("KK2/3") along with the inelastic scattering data of (Di 63) and (Bl 66a). The optical—model parameters are listed in Table V—l and V-2. The predictions of the collective model are normalized to the experimental total cross section. The value of the deformation parameter determined from the normalization was )6; =0.66 in agreement with previous results (Sa 67). The best agreement with the experimental data at both energies was obtained from the collective—model calculations. The agreement was particularly good at 40 MeV, where the shape was quite accurately predicted between 0° and 110°. However, the small backward peak observed at this energy Was not reproduced. The microscopic—model calculations are in generally 94 95 .90H0E090Q 0gp mo 05H0> .EDEHHQO. 059 0H G0>Hm thEDc 0coo00 0:9 .05H0> ESEHCHE 09H EOQM wmm smnp 000H >Q ©0000QOGH 0H WX p0£H £050 0>0£ G00 #09080909 03H 05H0> A900HH060V #00090H 0gp 0H C0>Hm,90oE:c AUQHLHV HmsHm 03H “0909080909 HHQQOIchm may pom An .mOm.Huop 0A or c0x0p 003 09HU0Q QEOHDOU 0:9 .OQ0N OH H0900 #00 0903 0Ho0p 0£p CH 900mm0 you 00 QOHSB 090H050Q0a £HQ0UIHH03 .A:I>H 0ostm 0000 00HUOH0 0OC00Q0000I>MQ0C0 may 5090 U0CHEQ0H00 0o03 090H0E0p0m 00099 AM 003.0 00.0 00.5 000.0 :0.H 00.5 050.0 0N.H 0H.0 000.0 00.H 0.0: 0.0: 000.0 0H.H 00.0 0H:.0 00.0 00.0 N5:.0 :0.H mH.5 0:0.0 :m.H 00.0 000.0 50.H 0.0: N.0N 000.0 mH.H 0H.0 a xevomm Aavoma Aax>mzaom> levee levee x>mzaaz A0000 A0000 Aamzva Aamzv m A0.0COHH0HSOH0O 0Hn09 96 .909060905 0:9 50 05H0> .E5EH950. 0:9 0H 50>H0 90:85: 050000 0:9 .05H0> E5EH5HE 09H 5099 00m 50:9 000H 5: 00000905H 0H 9x 90:9 :050 0>0: 500 909050905 0:9 05H0> N A900HH080V 900090H 0:9 0H 50>H0 900855 A09H:9V 909H9 0:9 a0909050905 9H:9OI5H50 0:9 905 A: .50m.Hu09 0: 09 50x09 003 05H009 :EOH500 0:9 .0900 O9 H0500 900 0903 0H:09 0:9 5H 900550 905 00 :OH:3 09090E0905 :9500IHH03 .H3I>H 0950H5 0000 00H0590 0050050500I>09050 0:9 609% 00GHE90900 0903 0909060905 000:9 AM 003.0 00.0 05.0 003.0 30.H 03.5 300.0 0N.H 00.3 050.0 00.H 0.03 0.03 000.0 0H.H 00.0 003.0 00.0 0H.0 303.0 30.H 00.0 050.0 00.H 55.N 0N0.0 50.H 0.00 N.0N 000.0 0H.H 00.0 a 900000 900000 A>mzoom> 90090 90090 A>mavaa 90000 “meme x>mzva A>mzo m H0.050H90H50H00 4035 0:9 CH 0005 0909050905 H0UOEIH00H950 H05c0:0 9me .NI> 0H009 .NI> 0:0 HI> 00H:09 5H 0090HH 090 0909050905 H00OEIH00H95O 0:9 H00 H50 97 >02 0.0mn5m 90 0900 :OH9000 00090 0H900H05H 0 . .0 380: 0 00_ 00 00. 00 00 _ _ _ x * x x ,/ *WW ** W TI n\~xxl.l «.27! 020950840 I .joolI :20: -5090 0.5. ed 250 + 3......800230 . as. 0.00 "am . as $93.00 1. Se p — _ — NH 1.9050 00_ 09 ON. 00 00 £000 Hm: >02 0.03 0:0 0:9 09 0990 <030 .HI> 095090 00 O _ _ _ _ .00 I * % 0 ea I _.O 0 m 1 0 0 . a I l .1 p w I _ U I0; 0- e/. €9.in /// I «.2IIIW0205538/1/ O— .JJooII. x) $2025.10 0:5 .50 230 w o “I .stdaiga. . >22 000.0 $39.00 9a 3 0e. _ 5 _ _ 5 CO. v o 98 poorer agreement with experiment. On the other hand, these models have no free parameters (all parameters are determined from nucleon-nucleon scattering data). In particular, the absolute magnitude of the cross section is predicted by theory so that the curves are not arbitrarily normalized to the total cross section as is the case for the collective model. It is interesting to note that the shapes predicted by the Kallio—Kolltveit interaction are similar to the collective-model predictions at both energies, and that the impulse-approximation prediction agrees somewhat better with the experimental data at the higher energy. V.A.Z. Inelastic Asymmetries The inelastic asymmetries calculated with the three models are shown in Figures V—2 and V—3, along with the experimental data of (Cr 66) and (B1 66a). Both microscopic— model calculations include the contributions of the s=l amplitude arising from the spin—dependent part of the interaction potential (see Sec.II.C.2). The collective model in which the spin—orbit part of the optical potential is not deformed does not lead to such an amplitude. The agreement obtained with the experimental inelastic asYmmetry data was at best only qualitative even in the collective model. The phase predictions in this model are reasonably good at both energies, but the calculated magnitudes are far too small. The impulse—apprOXimation Predictions are again quite different from those of the h‘“—"Q ’ . 99 om , r F I l '2C(p,p') '2C*(4.44) _ Ep= 26.2 MeV "‘ >_ §spm FLIP (THIS EXERIMENT) ./' tomb / \ a ___. -’ ‘\ m —-COLL. / \ 4 L— CALCULATIONS ---HJIA /' ,/ ' _ m m-szxa - \\ o l// . a: p aom— ' g g i .1 S u. r— é ‘ z . 6'. x? .. '. / 9/27, {— 00 T «18 F- 4 CRAIG,et. al. +0.4 *- EE __ <1 ,./°‘v'"'\. 0 ““r L -04 _. 1 l l l l O 30 6O 9O \20 \SO \80 9cm. (deg) Figure V-2. DWBA fits to the 12C spin flip, and to the inelastic asymmetry data (Cr 86) , at Ep=26.2 MeV. SPIN FLIP PROBABILITY MG) 100 0.40 1 * 7 I I 1" '2c(p,p') '2C (4.44) ~ Ep=4OMeV V in 1. 9 § SPIN FLIP (ms EXPERIMENT) [ 030F- J a . 1P 0 {—COLL. muons —-— HJIA L —-—KK2/3 i E L — . 0.20 E i i 010 ‘i 00 +08 +04 0 -0.4 - 1 l l l_ .L 0 30 60 90 \20 \50 \80 0cm§deg3 1’2 , Figure V—3. DWBA fits to the C spin flip, and to the inelastic asymmetry data (Bl 66a), at Ep=l+0.0 MeV. -1- ‘v' 9'; '6 «M nxv 1| c aka -1 Jud 101 collective model and are also in poor agreement with experiment. The quality of the predictions obtained in this model deteriorated at the higher energy where the cross section predictions, Figure V—l, improved. Finally, it should be noted that the predictions of the Kallio—Kollveit interaction again resemble those of the collective model. V.A.3. Spin-Flip Probability The spin—flip probabilities determined in this experiment are also shown in Figures V—2 and V—3. The average run time per datum point was approximately 30 minutes. As mentioned previously (Sec.III.D.2) there is an uncertainty in the absolute normalization of :9% at 26 MeV and 1.11495 at 40 MeV. The data exhibit the characteristic backward peak of approximately 30% at 140° which has been observed at lower energies (Sc 6”) and for other nuclei (Gi 68,Ee 68). The experimental total spin—flip probability, which is defined by: (Na) {5219’ 4'“— '4 5.1%: (9’ “7’ SF (v.1) ,0 where $3 (9) is the differential cross section and F(9) is the Spin-flip probability at center—of-momentum angle 69, is given in Table V—3. Note that spin~flip events constitute only a very small fraction (”VG%) of the total inelastic cross section. The spin—flip predictions of the three models ......n “no .2}!- ~ ~ _ - y__~_ )mIF- - ..._ .-“Ai.~_>’t::_. 102 am . I . 0.0m 000 000.0 000.0 090.0 0:0 0+000 0 . - 000.0 0.0.2 000 0000.0 0009.0 0000.0 0000 0+0 . . - . 0.00 009 0030.0 0000.0 0000 0 0000 0+00m0 0 .il . 0zvmm 9005090mxm 09x0000009 009000000000 900000000000 0sfim> 00020002 A> . .hpfiafln0009m mfiawlcwmw H0909 .ml> 0HQOH 103 (Figures V—2 and V—3) are in semi—quantitative agreement with the experimental data. The largest discrepancies occur at the forward angles where the spin—flip probability is consistently over—estimated. The predicted total spin—flip probability is approximately four times the measured value (see Table V—3). It is interesting that the collective model, which contains no s=l amplitude, predicts a spin-flip probability in reasonable agreement with the experimental data. We conclude that the observed spin flip is almost entirely due to the distortions introduced into the entrance and exit elastic-channel wave functions by the spin—orbit term in the optical potential. This implies that if any meaningful information regarding the s=l part of the inelastic interaction is to be obtained from spin—flip data, the experiment must be performed for nuclei having very well determined optical—model parameters so that the effects of the spin-orbit distortion can be separated from those of the 8:1 amplitude of the inelastic interaction. A series of calculations has been performed in which the parameters of the spin-orbit term in the optical potential were systematically varied in an attempt to determine the sensitivity of the spin—flip predictions to these parameters. First, we determined the range over which the parameters could be varied such that X; for the fits to the elastic data increased by less than 25% (see Sec.IV.C). The limits of this range appear in Tables V—l and V—2 for each of the 104 parameters. Distorted-wave calculations were then made using the upper or lower limits for one of the parameters while fixing the remaining parameters at their Optimum values. In each case, the form factors given by the impulse approximation were used. The results of these calculations at 26.2 MeV appear in Figures V—H and V—S. It appears that the spin-flip predictions are somewhat more sensitive than the inelastic asymmetries to the spin-orbit parameters. In fact, it should be possible to determine the spin-orbit term in the optical potential from spin-flip data in those cases for which a polarized beam is unavailable (Pa 68). A major difficulty is that it is not practical to program an automatic search routine for DWBA calculations. In the same spirit, a number of calculations were performed in an attempt to determine the effect of the 8:1 amplitude on the predictions of the microscopic model, again using impulse—approximation form factors. Two types of calculations were performed. In the first case, the optimum optical-model parameters of Tables V—l and V-2 were used, but the 5:1 amplitude was set equal to zero. In the second type of calculation the s=l amplitude was that predicted by the impulse approximation and the spin-orbit depth VSO was set equal to zero. The results of these calculations at 26.2 MeV also appear in Figure V-5. It is clear that the 8:1 amplitude has only a small effect on the spin—flip and asymmetry predictions. The predicted spin flip is increased by an amount which is almost independent of angle so that '105 30 l W W ] I2 . I2 5* 0030) C (4.44) Ep= 26.2Mev _ a. 3 20 - U. .3 ........ - a 0 o\° l0 0 l 4 L l j r I l l +0.5 r _v’°+ 25:0 .//’- \r.‘ - CALCULATIONS "“402“ . \\ -----R,°+25% \ - ----R,°-25% - 5 00 my. _ V - ___________ '\ // 0; N‘\ ///‘/ 05 1 . ' a . 0 3o 60 90 l20 :50 :80 emfideg.) Figure v-u. Dependence of the spin—flip and inelastic asymmetry predictions on the spin-orbit optical parameters. The notation ('125%') refers to the upper and lower limits for the parameters listed in Tables V—l and V—2. % SPIN FLIP AIS) 106 30 I2 . l ”-\ _ c(p.p) 2C"‘(4.44) / \ - - / h“ Ep-26.2 MeV /. \.\.\ 20 IO ......... 76:”w’v-‘u- ..-------..--------------\\\\ _,,_§‘ 1 /,- ~ .......... \ O ,-/' n I 4 L I 1 r 1 I I +0.5 P — As; 25% °\ - CALCULATIONS —“ A“,- 25% ----- Vso = 0.0 0.0 """ _’_ j» -------------- ----— r— d ‘0 5 L l 1 1 1 ' 30 60 90 I20 I50 I80 SQmIdeg.) Figure V-S. Dependence of the spin-flip and inelastic asymmetry predictions on the spin—orbit optical parameters, and on the 8:1 part of the microscopic-model interaction. I \ I °( .- 6 I o 107 the greatest differences occur at the forward angles, where the spin flip is smallest. No definite conclusions regarding the spin—dependent part of the inelastic interaction can be obtained from these results. The addition of an s=l amplitude to the microscopic- model form factors seemed to make the agreement with the experimental data worse, in that it increased the predicted spin flip at the forward angles where it was already too large. However, in View of the inability of-any of the models to reproduce the inelastic asymmetries, and considering the fact that Optical-model parameters which adequately fit all of the elastic data could not be found, it would seem that the difficulty lies in the failure of the optical model for nuclei as light as 12C. 12 * V.B. 0Sn(p,p')1208n (l.l7) V.B.l. Differential Cross Sections The differential cross section predictions of the three models for 30 MeV inelastic proton scattering from the first 2+ state of 1203n appear in Figures V—B to V—8, along with the experimental data of (Ri Ska). Calculations were performed for all of the sets of optical—model parameters listed in Table IV—2, but the resulting predictions were very similar so that only two of them are shown for each model. The collective—model predictions (Figure V—B) are in 108 no.0 r I r T 1 [I '205n(p,p')'2°5n*(l.l7) I : I Ep 30Mev / 0. o RIDLEY,et.oI. A / :73 K . cou. MODEL I— . \ ' B IO .0 —|B(fi2=0.ll8I ‘1 E . ---2c (32:0. '23 3 \V B t - I: \ '0 . ‘V‘ \ O.| - d A C 0.0: 1 1 J l l 0 3o 60 90 120 I50 I80 , 9cm(deg.) Figure V—B. Collective—model DWBA fits to the Sn inelastic . cross section data (Ri Ska) at 30 MeV. The identification numbers 1B and 2C refer to the optical-model parameter sets of Table IV-2. The deformation parameter is also given. 109 '00 a I I I I I / / \\ 'ZOSn (p. p') IZOSnW I. I7) / Ep=30MeV \ o RIDLEY. u. ol. L. . " ~ g .0 _. o.“ 3&1th APPROX. _ E ---2c g .’ Ir\\ \ .. I \ '2 . '0 . \ \ . ‘ , \ 0.| *- . ‘ I l l l L 1 00' O 30 60 90 I20 I50 I80 9cm (deg) Figure V-7. Microscopic—model DWBA fits to the 120 Sn inelastic cross section data, using impulse-approximation ‘~form factors. llO '00 .I I I I I r/ \ '2°3n(p,p')'2°5n*(I.I7I Ep=30MeV ORIDLEY,et.o|. t 33.- g ..o— __r§R°E 1 E ---2c ~—a \V .‘ g .. o " \ \ / \\ 3 .. «\ ’ \ o . \ OJ — . \r I I 00' l l l‘ l I * o 30 60 90 I20 I50 I80 . Figure V—8. Microscopic—model DWBA fits to the 120Sn inelastic cross section data, using Kallio—Kolltveit form factors. 111 very good agreement with the experimental cross section data. The values obtained for the deformation parameter Pt (Figure V—6) compare favorably with previous results (Ri 6Ha,Fu 68). Furthermore, the calculated value for the reduced transition strength B(E2) is in good agreement with gamma-ray measurements (St 58), and with theoretical predictions (Ra 67) as indicated in Table V—H. The microscopic—model predictions (Figures V—7 and V-8) are also in good agreement with the experimental data. As mentioned previously (Sec.II.C.3), these calculations were performed using the Yoshida wave functions (Yo 62) which include the effects of quasi—particle excitations from the closed neutron and proton cores as well as in the unfilled neutron shells (the 'nuclear cloud'). The results indicate that these wave functions give an adequate description of the l2OSn nucleus. V.B.2. §pin Flip The predicted spin-flip probabilities appear in Figures V-9 to V-ll, along with the data from this experiment. Each datum point represents an average run time of about two hours. The absolute normalization is uncertain by :5% due to the uncertainty in the efficiency of the gamma-ray detector (Sec.III.D.2). The experimental and theoretical total spin— flip probabilities appear in Table V—l. In contrast to the case for 12 C, the theory here under—predicts the total spin flip by about a factor of three. The theoretical predictions of all the models are very 112 . h A m omv QOflpoHpmhm Hwoapmsoofi . H .Amm pmv pcoEmQSmmmE hos MES A0 .m Capoppmom cowosm oapmmHmc an H Am m m oaxmmw. III 2 m m H :mNmmonAHH.oHom.mv m m oax . 2 m m Amm o:om.mv + SP ofimmvm mxm A Ammvm oxm An A Ammvm m .mcoapoapmsm Hmoa . . .pmsom msfisoppmom oawmwammw mmwomm paw mpcmamssmMmE mmsumaamm . 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The backward peak, which has previously been associated with the effects of distortions introduced into the elastic-channel wave functions by the spin-orbit term in the optical potential (Sec.V.A.3), is well reproduced in all cases. It is particularly interesting to note that the initial rapid rise in the spin~f1ip probability, which occurs at a center-ofvmomentum scattering angle of about 120° for the 12C data at both 26.2 and ”0.0 MeV and at about 130° for 58Ni spin-flip data at 15 and 20 MeV (Be 68,Ko 67a), occurs here at approximately 140°. This may indicate a systematic behavior in this feature of the distribution as a function of the mass number of the target. The location of this 'edge' is accurately predicted by the DWBA calculations. The situation is quite different for angles away from the peak of the distribution. The spin-flip predictions of all the models are significantly smaller than the experimental data in this case. In particular, the average value for all the experimental points between 30° and 135° is 0.103:0.020. The average predicted spin—flip probability over this angular range is about 0.025. Even if it is assumed that the acceptance—angle correction (Sec.III.D.B) takes on its maximum possible value at all angles, the experimental average is only reduced to 0.09H:0.020. Therefore, the discrepancy cannot be accounted for by the uncertainty in this correction. In addition, a comparison of the collective- model and microsc0pic-mode1 predictions shows that the addition of an s=l amplitude due to the spin—spin term in the .-..geiu-‘u-n- 3.1. v'. 117 two-body scattering (Sec.II.C.2) does not improve the agreement between theory and experiment. It happens that the various single-particle contributions to the form factor (Sec.II.C.3) add in phase for the 8:0 part of the interaction but cancel almost completely for the 8:1 part. As a result the 8:1 amplitude makes little or no contribution to the predicted spin flip. Calculations were performed using the complete impulse—approximation form factor (8:0 and 3:1) with the spin—orbit well depth in the optical-model potential set equal to zero. The predicted spin flip was essentially zero as expected from the comparison to the predictions of the collective model, which does not contain an s=l amplitude. It is possible that the forward—angle discrepancy may be removed by calculations which include some terms in the two-body scattering amplitude that have been neglected in this discussion. In particular, the two-body tensor and spin-orbit forces may make significant contributions to the predicted spin flip. The single—particle contributions from the latter, at least, are expected to add in phase (Mc 69a). It is particularly encouraging in this respect that collective-model calculations in which the spin—orbit term in the optical potential is deformed tend to predict larger spin-flip probabilities at the forward angles. (Sh 68,Gi 68). A microscopic—model calculation which includes the two—body tensor and spin—orbit forces should display the same behavior (Mc 69a). “JW’ r.‘.__-~_-u_--I' r . - CHAPTER VI SUMMARY AND CONCLUSIONS The spin—flip probability for protons inelastically scattered from the first 2+ state in 12C and l208n has been measured at incident proton energies of 26.2 and ”0.0 MeV for 120, and 30.0 MeV for leSn. The data display the characteristic backward peak which has been observed at lower energies (Sc 6H,Ko 67a) and for other nuclei (Ko 67a, Be 68,Gi 68). The magnitude of this peak is about 0.30 for 12C and 0.50 for l20Sn, and the location of its rapidly rising edge seems to be correlated with the target mass number. The total spin—flip probability is approximately 12C and 0.08 for 120Sn. 0.03 for The theoretical analyses of the data in the framework of the DWBA are in semi—quantitative agreement with experiment at the peak of the distribution. The most serious failure in this regard occurs for the 12C data at 40.0 MeV, when the predicted peak spin flip is only about 0.20 compared to the measured value of 0.30. However, larger differences are observed for the forward angle data. In the case of 120 spin flip, these discrepancies are of such a nature that no definite conclusions may be reached regarding the spin— dependent part of the inelastic interaction. The addition 118 '~¢J.'D'V __.o....—.--.a_(.(1v..k~‘“ .' ' - ’. 119 of an s=l amplitude in this case made the agreement between experiment and theory somewhat worse, in that it significantly increased the predicted spin flip at the forward angles where it was already too large. However, these discrepancies are in large part bound up with the general failure of the Optical model to give an adequate representation of a nucleus as light as 12C. The situation is somewhat different with respect to the 120Sn forward angle data. In this case, the Spin flip is Enggrfpredicted by about a factor of three. The remarkably good fits obtained from the optical model for the elastic cross section and polarization, and the fact that the inelastic cross section and the backward peak in the spin- flip distribution are very well reproduced in the DWBA calculations, lead one to believe that the calculated contribution to the spin-flip probability from the elastic— channel wave functions is essentially correct. In addition, there is some evidence that a more adequate treatment of the spin—dependence of the inelastic interaction will lead to an increase in the predicted spin flip at the forward angles (Gi 68,Ko 67a). This suggests that further DWBA calculations should be performed including those spin—dependent parts of the two-body scattering amplitude which have been neglected here. . The present experiment was directed toward the determination of the type of information about spin—dependent nucleon—nucleus forces which can be obtained from spin-flip data. Two general conclusions can be reached in this regard 120 from the previous discussion. First of all, the prominent backward peak which is characteristic of all the data presented has been associated with the distortions introduced into the elastic-channel wave functions by the spin-orbit term in the optical model potential. Therefore, this feature of the distribution may be used to determine the spin-orbit parameters when elastic polarization data are unavailable (Pa 68), and it can provide supplementary information in those cases for which the scattering of polarized protons has been measured. Secondly, it seems from the l208n data that meaningful information concerning the spin dependence of the inelastic part of the interaction can be obtained by careful measurements in the forward direction for those cases in which the optical model parameters are reasonably well determined. Two general types of spin—flip experiments are suggested by the results of the present study. First, 3He inelastic scattering and spin flip could be investigated, leading to a determination of the spin—orbit parameters of the optical model (Pa 68). Secondly, it would be of great interest to have accurate forward angle data for a set of nuclei having essentially the same optical—model parameters and different detailed structure (such as the even-even isotopes of Sn, Cd, and Te) to investigate the dependence of the spin—flip probability on the nuclear wave functions and the two—body scattering parameters. This latter investigation should yield information on the spin dependence of the inelastic interaction which is difficult to obtain in any other way. APPENDIX A THE BOHR THEOREM We wish to investigate the effect of a reflection in the scattering plane on a two—body scattering system which conserves total angular momentum and parity. Such a reflection may be obtained by a rotation of 180° about the normal to the scattering plane, followed by a parity inversion (Figure A-l). Denote the reflection operator by &a,, the rotation operator by R, and the parity operator by P. Then: a] = PR1(H) (A.l) For a system with total angular momentum 31 .sA a(I-zln’ «EMITF 121(7) '1' 5” = 8’ (A.2) where MJ is the z—component of the total angular momentum. The reflection operator becomes: LMJW 61.: Fe (A.3) It is clear that the eigenvalue of this operator will be a constant of the motion for any system which conserves 121 . iii - ._ __‘-#_--l. ..lmL 0" ‘. n - . ,4 .1 ' .Qowmhm>cfl mwflsmm m >9 pmzoaaom mfiXMIN may Psonw coma mo GOHPMpOQ _ m ha AOCMHQ >va madam mcwpmpmem one cw coapomammg m mo COMVflmOQEOU .HI< mfidwflm 122 N N ZO_mmm>z_ N .Somd. Ewhm>m >._._mm m._.<._.om w._.= (A.H) Now, the total angular momentum J is composed of an orbital and a spin part: ..hA—L J = L + S (A.5) Similarly, the parity operator can be divided into an orbital and a nuclear parity: P : P P (A06) so that the expression for the reflection operator becomes: i(22+$%)fl (Hz-1:13,, 6’ (A.7) where l (s ) is the z—component of orbital (spin) angular z z momentum. Next expand the state ,fifin the spherical harmonics: N (a 1 “935“ 9.1.. “’ Y, (924’) (A.8) .1 (1,)contain all the other coordinates n 4 O O O describing the system, such as the spin eigenfunctions and where the functions 4? the radial dependence. Because of the initial choice of 121+ axes, the coordinate 9 is always equal to 90°. We have: mse ( ”’1: "g QIW->= P e, Z O 1d)Pe Y (E?) ‘ " My: ”'1" 1 ‘ I (A.9) (HTS £+~k "'1 P” e i Z (—-> fix} E (EN 33¢ C “I: m Now, the spherical harmonic n I (792447) vanishes unless 1 + m1 is an even integer. Therefore: L'n‘s (XIII?) = Re 2[#1) (A.10) A similar expansion can be made for the final state %; The detected particle is again in the scattering plane (by definition) so that the results are the same. Substituting into (A.H): irrs CW5 <fi/Fhe EIW> ‘-’- <53’Pne’ EWI> (A.1l) We have recovered the Bohr theorem, which simply states that the eigenvalue of the operator: L'Tl‘sZ &’ :: PM 9 (A.12) is a constant of the motion for a two—body scattering system which conserves parity and total angular momentum. APPENDIX B ACCEPTANCE-ANGLE CORRECTION The general angular correlation function W( 4a,,55.,¢;) for the de—excitation of the 2+ nuclear state to its 0+ ground state is (Sc 6”): M ’W‘. ( *( 2: Z I! ' m 5'”! ’W where X is a normalized vector spherical harmonic (Bl 52) 2m of order two, 11 represents the possible combination of spin orientations of the projectile in its initial and final state, P1, is the probability that a particular combination will occur, and am(zl) is the amplitude for exciting the mth magnetic sublevel with that combination. For proton scattering there are four such combinations: 1V =1: incident spin up, outgoing spin down ‘V =2 incident spin up, outgoing spin up (B.2) 1I=3z incident spin down, outgoing Spin up I’=H: incident spin down, outgoing spin down If the quantization axis is chosen along the normal to the 125 126 scattering plane, the Bohr theorem (Appendix A) requires that: a.,(v)=- o ”If ”H.” k “U‘ (13.3) Applying this restriction and evaluating the spherical harmonics: . I Iva—mums «mama; {Ia-1:111 +'°‘11’I:P{"‘"”"“””)] r . " I 5 + 9"; [Sent 9’ «5‘9, (E; [norm/1+ P9 IQaUU’ )) J ., z 4 5:" [( l—mVB’) (F19 sz")/l+ haunt} + P.‘ {Iqtcwl +14%“)! D] It- egu pr 167C [2““19, ”m 9-'7(P,f2¢{a1_(1)a1(11 } 16¢ + a: { at“) 0;:(3’ 8’ 7})1 (B. n) (P Enigma“) Cu i1’ -£r[2su~‘,1r9“’9 3n + P 0?. {a M “£3” 62.49,} £0 +Pq flaiafléu)a‘“°z ’1 100 WW (“I (1:008 })] QLQ [ “9(Pfisfama:m6 1'} MT 2‘“ 4w, ) +an‘iq‘f‘0a‘qfi'ue z] 127 Now, let: P (Q (“It + P (0 U0, s;‘= 1 o H O x 1 z I 51 .- I; {(a1till + 131(1)] }+ F: {loam} ””40” (B.5) 'L 51 = P, {/QJUI‘I- 1'11“”! }4 I; {law/‘4 Igthfiz} _ 5F...(vl awn/I - th'z/J e The quantity S1 is the probability that the m=:l magnetic substates of the 2+ level will be excited, i.e., the spin— flip probability. The next step is to evaluate (3.4) for S1 in terms of the measured angular correlation function W. For a detector at a small angle 9 =‘ E i , (B.H) leads to: S1: ¥W(¢PIO.JOO) - 6"(3$°+-$,— af $1) (3)) _ 3“- ]: 7:141) no: (2413+ P1_f11’)+ l,”"'°(“.*5~1 J (B.6) +1}: e" ET..“~’ Q" (zo,+ Fw‘”) + 1:" coacw,+F.-{‘J) A + (‘0) ... T100!) U’zhcprI-Fw‘W) + ’0 ~20!) ”(1% F0—1. J 4- C976") where: ‘J .(K) ) : .(K) _'ng PM” P. 128 The expression (B.6) must be averaged over the apertures of the proton and gamma-ray detectors. The first term is independent of 6 and is unchanged by the averaging procedure. The quantity 5W2; is the normalization constant which gives the spin-flip probability in terms of the gamma—perpendicular correlation function. The rest of (B.6) may be divided into 'direct' terms, proportional to 6" , and 'interference' terms, proportional to E‘cos( 20, + [$23.00) . When averaged over the apertures of the proton and gamma-ray detectors, these terms give the acceptance—angle corrections. In this development, the gamma detector is approximated by an 'equivalent' detector of zero thickness and the same intrinsic efficiency situated at the center of the actual cylindrical detector. The effect of the finite aperture of the proton detector is to define a set of scattering planes whose normals are tilted from the axis of symmetry of the gamma- ray detector (Figure B—l). Let: 6' = angle of tilt of the normal to a given scattering plane relative to the center of the gamma—ray detector. (B.7) 5:) = half—angle subtended by the gamma-ray detector. é ’ integration variable defined in Figure B-l. Then the edge of the gamma—ray detector referred to the point at which the normal to the scattering plane intersects the surface of the detector is described by the equation Of 129 DETECTOR GEOMETRY -—v—g' Y 6 63> I p x I‘ PROTON DETECTOR I I169 I I I I ‘z—CAMMA I‘ DETECTOR .v -2 NORMAL TO SCATTERING PLANE (25 --y NORMAL TO I SCATTERING PLANE I INTERSECTS HERE x GAMMA DETECTOR (VIEW ALONG Z -AX|S) Figure B-l. Detector geometry for the calculation of the acceptance-angle correction. 130 an off—center circle in polar coordinates: Ez- " (700,4 * W/SD'TGP‘5‘4‘4’ (B.8) Muv The average of the direct terms over the gamma-ray detector is proportional to: 1n Sr <6 1.) Sér‘J-nvr L Sgt-Ian’J J¢ = z 6 6 " 5m, "‘0 . o " (B 9) which reduces to: 1. 5'1 9 7- -= 6 + .D. .. '1 6.1:. The interference terms are somewhat more difficult because of the dependence on £2 -(¢+L;). In this case, the equation of the detector edge is written in the form: 1 1_ 1 (1) 63'0“» 4'6, ED C05 (U(mtx 5 m» .26, a: (B.10) 62f :- 69 I’ 60 Max Then, the average of the interference terms is: ether gu‘y .1. 5 5 , “9,14; J5, <5; cm (23 131 The expressions (B.9) and (B.ll) must now be averaged over the proton detector aperture. For a rectangular aperture, all values of G} are equally weighted. For a circular aperture of radius E , the weighting is W . The final results for the direct and interference terms are given in Table (B-l). The acceptance-angle corrections are determined by substituting these results into (B.6) using the fact that S0 + 81 + S2 = 1. For example, for a circular proton (gamma—ray) detector subtending an angle of 2.6 (ZGD ) radians the maximum and minimum values of the correction terms are given by: 6.3 60‘ .L E! -11 -3. l -113 510-5 MAX=fl ”+Y-64503)(31$1) 365; 9 1) (B.12) q 1 6 t J. 6 ) 1 3 E D - ... .— where it is assumed that the /%.(K) are independent J quantities. 132 Table B—1. Average of direct and interference terms over the acceptance angle of the gamma—ray and proton detectors. APERTURE a) TERM Direct Interference z ‘I 5: ED _I_ __ J5 51cc; F._(k) RECTANGULAR 3 * ‘2‘ “Po e1 “ D 9 I. 'l- I 6 _L ‘- 0‘) 6. Go - -- é “5 .'° — __ — J CIRCULAR I, + 2 c5! 60" ‘I F a) Refers to the proton-detector aperture. The quantity 5 equals half the height for a rectangular aperture, and is equal to the radius for a circular aperture. The gamma-ray detector is assumed to be circular, with radius 6 in all cases. 3 3 APPENDIX C DERIVATION OF THE FORM FACTOR IN THE QUASI-PARTICLE MODEL The form factor is proportional to the reduced matrix element: < HIV“), E‘JIIO> taken between the initial and final nuclear states which are to be described as quasi-particle states (Yo 62). Applying the Wigner—Eckart theorem (Me 65): MT <1" V1“. IMHO) .... (0.2) ..IMJ The single-particle Operator ya“; lei may be written in second—quantization notation (Ma 65): "I . ’5' . T T . =— z J M", J: ' (C.5) J-hI T' f C16) X [1%, K (.2 “Jam, “Ii-'4” + VI 4 “‘62", duh] + (terms which have zero matrix element between the ground state and the two quasi-particle excited states). Following Yoshida (YQ 62), the two quasi-particle operators are defined by: *- 'fl "1; . z °(;" PI”, M, ”It ”I 9 'v Af (J, Jot IM) :- (C.6) A (In. I“) 5... < J"‘”"""‘T”> 5339M, H In terms of these operators: ..M. z J4 J: (In: Emu.» (J’VI Ii) I ' 13' ... 154 J JJ (C. 7) xEV ’V A 0“,;an mam/“09”“) 135 A r. .I . . where J = 2.1+, . The matrix element (J '{15JIJ> is a radial integral of the interaction potential taken between shell model states which are assumed to be harmonic oscillator wave functions. Next the phonon creation operator is defined: 4- 7 + :r ""J .. _ I. -- Ar... :40] .. .. J JM _. (PM?) , . , J 9.74": 1J§[%’»A ('J‘ a) '6’. (C-8) I J where P and ¢ . are normalization coefficients (Yo 62). Then: J’Jv “0* ‘ I’M; A+(J J TMJ) = (-) <17”. 62“,] + ‘7’},- am, I I .T A (I'JJMI) a S‘jj Qa‘n +(-) (P. so that: V T _, 5 .I f Tlsj ‘lti -. JJ' J J I T (C.lO) . 4 . .1 U] m. w. . J R... + (phonon destruction terms). 136 The Q Operators satisfy the boson commutation relations: [‘9 , ‘22,“) = Sn' 84,”, ((3.11) In and the one-phonon excited states IJMJ7> are given by: ’- Imp - QM, Io> ((2.12) where IC£>is the ground state (phonon vacuum). Thus, the reduced matrix element (C.1) is: A A" , . (III v,” I1”. II0> = .5. J. I 211 H. Sherif and J.S. Blair, Phys. Letters 26B, 989 (1968). J.L. Snelgrove and E. Kashy, Nucl. Inst. Methods 5 , 153 (1966). P.H. Stelson and F.K. McGowan, Phys. Rev. 110, 989 (1958). W. Tobocman, Theory of Direct Nuclear Reactions 192 Phys. Phys. Phys. Phys. Phys. Phys. Soc. Tobocman, Phys. Rev. E. u. A100, 997 (1967). A95, A92, :82. A68, 1037 (1955). 1 (1969). 981 (1966). 1 (1967). 273 (1967). Schmidt, R.E. Brown, J.B. Gerhart, and Kolasinski, Nucl. Phys. (Oxford University Press, 1961). J.G. Valatin, Nuovo Cimento 1, 893 (1958). H. Yoshiki, Phys. Rev. 117, 773 (1960). s. Yoshida, Nucl. Phys. 39, 380 (1962). 118, 353 (1969). 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