C ELASTI' .IN 3.1. 2;. “WHAT 1:! , . . . imi 1} 5; :s‘ 31, N; he. I .y 5 . CH1 {w affix. ‘ Ed .1: 1.: 35.x... Ema. ‘9 IV. ,5fl f.” :r. 4/2. ELL y-Hoiw RR . 3.4}. .f. f. E: III ritr.>.yil r) .1}!!! . :1: RDTH {RIcHA . . .1 5.! .U , . [Mohnmif 3.325 (a... (If: .1 {I'VIV .I .. . )3.r.i.1!!. .15.? I... .n... HQ. l-« V. §.hl l..;~.‘1!'. .3r.....-1.......s....-. I. 53.5. . 3...,me nnfifixrfinflfi. x\ .WRVug... .. .1. ‘flm..§\| .Jliwmin (A! Libs. 4RY Midig .nStaUc ‘Jni=1§ky' “ ‘ This is to certify that the Z thesis entitled »?5‘A PROTON SPIN FLIP PROBABILITY IN INELASTIC SCATTERING l2OSn AND 1243n AT 30 MeV presented by ON Richard Harry Howell has been accepted towards fulfillment of the requirements for PhD Jegree in Phys :‘LC 8 @200” I V Major professor Date January 5, 1972 0-7639 I |IPIIIV ‘l ABSTRACT PROTON SPIN FLIP PROBABILITY IN INBLASTIC SCATTERING ON l2OSn AND l2uSn AT 30 MeV By Richard Harry Howell Proton spin—flip in the excitation of the first 2+ l2OSn and l2”Sn has been measured at 30 MeV states in using the (p,p'y) coincidence technique. The data are fit by the DNA collective model using the full Thomas spin—orbit coupling term and by the DWA microsc0pic model using detailed wave functions and a realistic interaction. These calculations are also compared to published proton angular distribution and asymmetry data on the first ex- cited state in 120Sn at 30 MeV. The effect Of including a realistic two body spin—orbit interaction was investi— gated with reSpect to these data. The effect of complex coupling was also investigated for these data and pub— lished cross section and asymmetry data on the first 208Pb and 58Ni. Imaginary form factors excited states of obtained from the collective model and from a phenomeno- logical microscopic prescription were used. It is con- cluded that an imaginary term in the form factor can be important. PROTON SPIN FLIP PROBABILITY IN INELASTIC SCATTERING ON l2OSn AND l2uSn AT 30 MeV by Richard Harry Howell A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1972 ACKNOWLEDGEMENTS As are many doctoral students at the MSU Cyclotron Laboratory, I am indebted to the entire Cyclotron staff for their ready assistance during the course of this work. I wish to thank Professor H. McManus, Dr. F. Petrovitch and G. R. Hammerstein for their assistance and encourage- ment in performing theoretical calculations and to R. Frick for assistance in running the codes. I am grateful to M. Savoie and R. Doering for the many nights spent assisting in the collection of the data. I am happy to thank Mrs. Mary Krueger for her cheer— ful assistance both in the computer room and in typing this dissertation. Finally and especially, I am indebted to Professor A. Galonsky. His advice, assistance, and encouragement were most valuable during my tenure at MSU. ii TABLE OF CONTENTS List of Tables List of Figures 1. Introduction 2. Nuclear Theory 2.1 Approximations in the Treatment of Scattering 2.2 Optical Model 2.3 DWA Basic Formalism 2.4 Methods of Calculating the Spin Flip, Polari- zation and Asymmetry in the DNA 2.5 The Collective Model 2.6 The Microscopic Model 3. Experimental Procedure 3.1 Cyclotron and EXperimental System 3.2 Detectors 3.3 Electronics 3.4 Data Reduction 4. Data Analysis 4.1 Inelastic Scattering Data 4.2 Elastic Scattering Data and Optical Model Parameters 4.3 Collective Model Calculations 4.4 Microscopic Model Calculations 5. Summary iii vi IO l2 13 15 19 24 27 32 41 56 57 60 70 86 6. Appendix List of References iv 88 98 LIST OF TABLES Clement and Baranger wave functions and occu- pations for 120Sn and l2”Sn 2+ states. 23 Optical model parameters. 59 Ground state charge density parameters. 91 WWWLUUU \10\U‘l .10 .11 .12 .13 LIST OF FIGURES Proton spin—flip on targets ranging from 12C to 6ONi at beam energies from 15 to 40 MeV. Experimental area of the Michigan State Uni- versity Cyclotron Laboratory. Side view of vacuum box housing the particle de— tector. Typical proton pulse height spectrum. Block diagram of the electronic system. Typical TAC output pulse height spectrum. Two parameter coincidence data. Gamma—ray singles and coincidence spectra with standard line shape. 120Sn spin—flip data without detector solid an— gle corrections. l2“Sn spin—flip data without detector solid an— gle corrections. 120 Sn spin-flip data with an average solid an— gle correction (q = 1). 12“Sn spin—flip data with an average solid an— gle correction (q = 1). l2OSn spin—flip data with maximum (q = w) and minimum (q = 0) solid angle corrections. l2“Sn spin-flip data with maximum (q = m) and minimum (q = 0) solid angle corrections. Collective model calculations of the 120Sn spin— flip including the deformed spin orbit with the BGOM. . . 124 . Collective model calculations of the Sn spin- flip including the deformed spin orbit with the BGOM. vi 25 3O 31 33 35 38 4O 48 49 51 52 53 54 61 62 JI'Jr-Ir-J: .10 .ll .12 .13 .14 .15 .16 .17 vii Collective model calculations of the 120Sn asym— metry including the deformed spin orbit with the BGOM. Collective model calculations of the l208n cross section including the deformed spin orbit with BGOM. Collective model calculations of the 120Sn spin— flip with Optical model parameters BG 70 #3. Collective model calculations of the 120Sn asym— metry with optical model parameters BG 70 #3. Collective model calculations of the 120Sn cross section with optical model parameters BG 70 #3. 120 Microscopic model Sn spin—flip calculations. Microscopic model l2uSn spin—flip calculations. Microscopic model 120Sn asymmetry calculations. 120 Microscopic model Sn cross section calcula- tions. Microscopic model 120Sn asymmetry calculations with a two body spin orbit force. MicrOSCOpic model 120Sn spin—flip calculations with a two body spin orbit force. Microscopic model 120Sn cross section calcula— tions with a two body spin orbit force. 120Sn asymmetry microscopic model calculations including complex coupling. 120Sn cross section microscopic model calcula— tions including complex coupling. l2OSn spin—flip microscopic model calculations including complex coupling. Real and imaginary form factors. 120Sn collective model asymmetry calculations. 120Sn microscopic calculations including complex coupling. 208Pb and 58Ni microscopic calculations inclu— ding complex coupling. 64 65 67 68 69 71 72 74 75 77 78 79 82 83 84 92 93 95 96 1. INTRODUCTION In order to learn about the spin—dependent part of the interaction in an inelastic scattering reaction there are various measurements possible. The angular distribu- tion of the differential cross section, asymmetry, polari— zation and in special cases the projectile spin flip may all be measured. The probability of a spin-flip event occurring may be measured through the particle—de-excitation gamma-ray angular correlation function with the gamma—ray detector fixed perpendicular to the scattering plane. Measure— ments of the angular distribution of the spin flip prob— ability of scattered protons have been reported on the lowest 2+ states of several even—even targets with mass numbers ranging from 12 to 64 and incident proton ener— gies ranging from 10 to 40 MeV. There are also spin flip data reported on some of these targets for the scattering of medium energy helions and deuterons. This report shows angular distributions for the pro- ton spin flip probability taken on the lowest 2+ states in 120Sn and 12“Sn at 30 MeV bombarding energy. The data were compared with calculations done in the Distor- ted Wave Approximation. Asymmetries and cross sections were calculated and compared with data on 120Sn taken elsewhere at the same energy (KA 70). The four measured quantities, cross section, asym— metry, polarization and spin flip, are related to the 1 2 set of partial cross sections corresponding to specific entering and exiting projectile spin projections along the normal to the scattering plane. These may be written: Cross section 0 = O++ + O+_ + O_+ + O__ Asymmetry 0A = O++ + O+_ — O_+ — O__ Polarization OP = O++ — O+_ + O_+ — O__ Spin flip 03 = 0+— + 0—+ (1.1) The symbol subscripts +- denote incoming projection + and exiting projection -. The cross section may be measured using an unpolarized beam by detecting the number of scattered particles in some solid angle. The cross section is: differential cross section = number of scat- tered particles / (number scattering centers x number of incoming particles x solid angle). The asymmetry is measured with a polarized beam by detecting the difference in the number of particles scattered into the same solbiangle at scattering angles + and - 6. The difference is then normalized to the sum of the scattered counts to obtain the asymmetry. The polarization is measured with an un- polarized beam by measuring the difference between the number of spin up and spin down particles scattered into some solid angle. This difference normalized to the sum of the scatteredcxnnnxsis the polarization. 3 The measurement of the Spin-flip probability through the proton gamma-ray correlation function is deduced with the aid of the Bohr Theorem (BO 59). This theorem is model independent, depending only on reflection symmetry in the reaction plane for its derivation. It may be simply stated AMS + AMJ = + / — as the change in parity in the reaction is even/odd, AMS (AMJ) is the change in the pro- jection of the projectile (target) spin along an axis normal to the scattering plane, the Z axis. In the case of a J = 0 initial target state, information about the be- havior of the projectile spin projection during the re- action is retained in the population of the sublevels in the excited target state. During the radiative decay, only IAMJI = 1 transitions are non-zero along the Z axis + + and 0+ to 2 inelastic scat— (SH 70). Thus for 0+ to l tering, a projectile spin—flip will always produce a de— excitation gamma-ray radiation pattern which is clearly separated along the Z axis from the radiation from non— spin-flip events. For higher spins IAMI i 3 channels are also Open for de—excitation Of sublevels populated by spin—flip. The de—excitation gamma—ray radiation pat— terns from some excited state contain the most useful in- formation about the excited substates for decays to a J = 0 ground state where the substate quantum numbers re- tain a unique correspondence with the magnetic quantum numbers of the transition Operator. Thus all measure- ments reported to date have been done on the first 4 excited 2+ state of an even-even target nucleus. Also, lower mass targets are preferred on experimental grounds, as the ratio Of the gamma-ray yield from the first ex- cited state to the total gamma-ray yield is higher and the energy Of the state is generally high. The lightest 12 isotope studied has been C at 10.3 MeV (SC 64), 12 — 20 MeV (KO 69B), 15.9 and 17.5 MeV (WI 71), and 26.2 and 40 MeV (KO 69A). In the s-d shell, Mg2)4 has been done at 28 10.3 MeV (SC 64), Si at 30 and no MeV (GI 68) and S32 at 15.5 and 17.9 MeV (WI 71). Heavier isotopes studied 56 include Pe5Ll at 10 MeV (AH 70); Fe5” and Fe at 19.6 MeV (HE 69); Cr50 and Cr52 at 12 MeV (SW 71) and a great amount of data due to the University of Washington group 58 at 10.3 MeV (SC 64), 9.25 - 20 MeV (KO 69B) and 20 MeV (BE 71); N160 at 10 MeV (AH 70); Ni60 and N16” at on Ni 10.5 and 14 MeV (KO 69B). Thus the presentation of data on Sn120 and Snl2u at 30 MeV considerably extends the range in mass of isotopes studied. Data have been taken with other beams. He3 spin flip is reported on 012 at 22.5 MeV (PA 68). Deuteron half 2u,26 32S spin flip (AMS i1) has been reported for M8, , L‘8’50Ti, 58’60Ni at 11.8 MeV (H1 70). With the exception of some data taken at energies for which compound nuclear effects are important, all the above data show the same general features. First, in every case there is a large peak from .3 to .4 in magni— tude which dominates the spin flip probability angular 5 distribution. This peak occurs at back angles with a maximum at around 150° proton scattering angle. Collec- tive model calculations which normally have not included any spin transfer in the nuclear interaction fit this back angle peak in most cases. In all cases a back angle peak is predicted. The collective model sometimes fails in fitting the height of the peak. This is usually as- sociated with a failure of the optical model in fitting the elastic scattering and polarization data. A second general feature of these data is a uniformly low, less than .1, and smoothly varying angular distribution at angles forward of around 100° proton scattering angle with a small peak at about 75°. Exceptions to this tendency do occur. In some reported data, the second peak is seen in the general vicinity of 90°. This peak with a magni- tude of about .2 is not as large as the back angle peak and is not always predicted in collective model calcula- tions. The collective model calculations are only depen— dent On the values of the optical model parameters. The general success of the collective model in fitting spin-flip probabilities over such a wide range of target mass and energy suggests that the dominant processes pro- ducing a spin-flip are dependent on the Optical (dis— torted wave) channels in the spin-flip scattering. An attempt has been made to use this property to determine a value for the optical spin—orbit well depth in 3He scattering from 012 (PA 68). Proton scattering from C12 and S32 have been studied and the Optical model parameters adjusted in an attempt to fit both spin flip and elastic data (WI 71). In Figurelxl thecollected data for all targets at all energies for which direct reactions are dominant are displayed. The uniform character Of the spin flip angular distribution is easily seen in this figure. The spin flip data on 120Sn and l2“Sn reported in this thesis are typical of those data just described. The increase in the mass of these targets over that of pre- vious targets does not significantly change the character Of the data. A small isotopic effect appears at angles forward of 75° proton scattering angle. In this region the values from 120Sn are consistently higher than those l2“Sn. DWA collective model calculations including from a full Thomas distorted spin orbit term (SH 68) fit these data well. Varying the spin orbit deformation from 0 to twice the deformation of the central well did not greatly affect the quality of these fins.Comparisons with pub— lished asymmetry data were more sensitive to the spin orbit deformation. They indicate that the spin orbit and central deformations should be about the same. Micro- scopic calculations using the tin wave functions of D. M. Clement and E. Baranger (CL 68) and the Kallio—Kolltveit force also fit the spin flip well. The real, central Kallio—Kolltveit force predicts the asymmetry badly. An PROTON SPIN FLIP FROM. THE FIRST .4 _ 2+ STATE OF VARIOUS TARGETS a h) T M I h—o—q 0-0-0 # Fit-O I-O-l i—Q-‘d O—o—I l L . 3’” I ' III .I I 1‘ [Ilj 1 I 1111,19 O .1111! l l I O 60 I20 I80 PROTON SCATTERING ANGLE (DEG) Figure 1.1 Proton spin-flip on tar— gets ranging from 12C to 60Ni at beam energies from 15 to 40 MeV. 8 imaginary contribution was estimated. The quality of the asymmetry fit was greatly improved by this addition. The fit to the cross section was also somewhat improved. 2. NUCLEAR THEORY 2.1 Approximations in the Treatment of Scattering Reactions which occur in a time interval comparable to the transit time across a nucleus are often thought of as direct reactions. The theoretical treatment of these reactions in the plane wave approximation is out— lined by Tobocman (TO 61). In this approximation, the interaction potential is treated as a perturbation and the incident and exiting particle wave functions are plane waves. A more complicated approximation separates the elastic scattering interaction potential from the total interaction potential. The incoming and exiting channels are then described by the wave functions for elastic scattering. This is the distorted wave approxi- mation (DWA). The algebra of this approximation has been discussed by G. R. Satchler (SA 64). A brief discription of the DWA, methods for treating polariza— tion phenomena in the DWA and the use of some nuclear models in the DWA are found in the following sections. DWA calculations were performed with DWMAIN, a code written by T. Tamura and R. M.Haybron at ORNL and mod— ified at M.S.U. and a collective model DWA code written by H. Sherif at the University of Washington 10 2.2 Optical Model In order to accomplish the more sophisticated cal- culations Of the DWA prescription, elastic scattering wave functions must be calculated which are a more ac— curate description of the scattering than that of plane waves. To describe elastic scattering and polarization of f protons, a phenomenological scattering potential has been developed. The strength and shape of this potential has been parameterized. Searches are made on the po- tential parameters attempting to minimize the chi— squared values of calculated and experimental elastic cross sections and polarizations. The experience of a great many researchers in applying this model to a wide variety of elastic scat- tering data has resulted in a successful potential form which has become accepted in describing proton elastic scattering. The accepted optical model scattering potential in use for the elastic scattering of protons at this time is U(r) = V(r) + iW(r) (2.1) where V(r) = V6(r) — VR f(r, RR’ aR) h2++ld + Vso( mflc) O°£ F'EF f(P’Rso’ aso) d + wSF 4aI 5—1"— f(I‘, RI, 8.1) The Coulomb potential, V0, is normally taken to be the potential between a point charge, ez, and a sphere of uniform charge, Ze and radius RC. Thus 2 2 Zze r 2R0 [3 - RC :1 9 r : RC VC z Zze2 r , r > RC (2.2) The radial functions f(r, R, a) are of the Woods— Saxon (Fermi) form f(r, R, a) = [1 + exp (r - R) / a1"1 (2.3) The nuclear radius R is further factored into RX = rXAl/3 where A is the atomic mass number of the nucleus. The notation used here is consistent with that of F. D. Becchetti and G. W. Greenless (BE 69). The distorted waves used in DWA calculations of in— elastic proton scattering are obtained from a potential of this form. Also small deformations from the spherical shape of this potential are expected in a macroscopic picture of an excited nuclear state. Thus in the macro— scopic (collective) model of inelastic scattering, the scattering interaction potential may be deduced from the optical model potential. 12 2.3 DWA Basic Formalism Using the notation A(a,b)B, the differential cross section is written 2 |T|2 g9 _ “a“b Kg MAMBmamb (2 u) — U do (2nh2)2 ka (2JA¥I)Z2JéIl) The mds are the reduced masses and the k's are the momenta. The transition amplitude, T, may be written _ —> —> —*—>+ ++—> T — fdra fdrb Xb (kbrb)Xa (kara) (2.5) where acts as an effective interaction producing the transition between the elastic states. The separation vectors Pa (Pb) are the relative coordinates of particles a and A (b and B). The transition amplitude may also be written in terms of reduced amplitudes, 2mmamb st _ t _ Am m T — £§j JBSjb a (2.6) where 2mm m b a .—£f-1 . . = l E <2sm' m'—m' m—m +m > BSJ J m'm'm' ’ a b '3 b a a b , sb‘mb l__l I_I _ x ( ) H" —> —> — —>—> +-—> x fdra fdrb Xmé (kbrb) Gisj,m'(rb’ra) + X' +» m ma (kara) and 3 = /2j+l . (2.7) 13 -> -> stj,n1(rb’ra)is the radial form factor and contains all information about the radial part Of the interaction. The n + m'm X (XQIm)are the distorted waves for the incoming (outgoing) particle. Part of the dependence of the dis— torted waves is on the projection quantum numbers of the, projectile as a result of the L '8 force in the optical potential. This dependence allows spin flip, polarization and asymmetry to be predicted in the absence of any spin dependent terms in the form factor. The integral in equation 2.7 is over six dimensions and is time consuming to evaluate. To simplify this integral Pa and Pb may be taken to be parallel. This is obtained by assuming that particle b emerges from the location at which particle a is adsorbed. Algebraically this condition is Eb = (MA / MB) Fa. This is the zero range approximation. In this approximation, the parity change in the reaction is just (—)2 where l is the trans- ferred orbital angular momentum. Particle exchange may be included in an approximate manner in the zero range ap— proximation (PE 69), (PE 71). 2.4 Methods of Calculating the Spin Flip, Polarization and Asymmetry in the DWA There are two viewpoints from which spin dependent quantities may be calculated. The first requires the de- velopment of a density matrix, 9, for the interaction. With the density matrix, formulae may be developed for the polarization and spin flip. A detailed description of 14 angular correlations calculated in the density matrix formalism is found in reference (RY 70). The second method divides the cross section into (2sa+1) x (2sb+l) partial cross sections a , where ma and m are the mamb b particle spin projections taken with respect to a Z-axis perpendicular to the scattering plane, i.e., Z is along + k3 x Rb. For spin 1/2 particles, the spin flip, asymmetry _ and polarization are simple sums Of these partial cross sections. These two approaches can be shown to reduce to one another. Since the cm m 's can in principle_be separately a b measured, it is appealing to the experimentalist to use them to calculate the other quantities. The defining formulae for 5a = 3b = 1/2 are 0 = 0++ + O__ + O+f + O_+ OA = 0++ + O+_ - O_+ - O__ OP = O++ - O+_ + O_+ - O__ °8 = °+- + °—+ (2.8) If the Z—axis of quantization is taken perpendicular to the scattering plane (in the direction Ea x Kb), these partial cross sections may be calculated by performing the sum over MA and MB in equation 2.4 while keeping the projectile spin projections distinct. Most formulations of the DWA algebra choose the Z—axis to be along the direction of the incident projectile momentum, Ea' This 15 choice greatly simplifies the algebra used in calculating the reduced amplitudes. However, the reduced amplitudes calculated in the coordinate frame with Z along fia may be rotated into the coordinate frame with the Z axis along Ra x Rb. The Om mb's may then be calculated directly. a The form of this rotation is (SA 64) Imebma AHHm'mé st (a1) = Z st (a2) .* s s JIR>fl° (R>,Da (R) I I ! OD u IJ 21 mbmb 21 mama 21 (2.9) where u = m + ma — mb and R21 represents the set of Euler angles (BR 68) necessary to rotate coordinates a2 into coincidence with coordinates a1. Theflj, (R) are u u the usual rotation matrices. The set of Euler angles (a, B, y) = R21, which perform the rotation of Z along Ea t z 1 I I ' R - 2 2 o a ong ka x kb , is 21 - (-fl/ , —fl/ , 0). The code DWMAIN has been modified to calculate the rotated re- duced transition amplitudes. From these, the partial cross sections, polarization, asymmetry, and spin flip are all obtained. 2.5 The Collective Model The interaction potential for the collective model is derived from considering a deformation of the spherical nuclear potential well. The spherical potential well chosen is the one which gives the correct elastic scat- tering, i.e., the Optical potential. A complete 16 treatment of the deformation Of the full Optical poten- tial, including the spin orbit term, has been done by H. Sherif (SH 67). His treatment is outlined here. The T matrix for scattering from a 0+ ground state to some state Of spin J, parity (-)J is T = 2 (X7' 3 3 ' U 00 l/2 ' > mamb mbmb (kara)| (2.10) 8.8. where AU is the first order deformation of the Optical potential. It is conventional to write this as a sum of the terms in the optical potential AU = AUC + AUR + AUi + AUSO (2.11) The central parts of AU are Obtained by expanding the radial parameters in the density function f(r, R, a) so that R + R + d(r), and f(r, R + a,a) = f(r, R, a) + a(r) gg where r is the angular coordinates. The central terms of the Optical potential become - A 3 AUr + AUi - —d(r) (VR 3R; f(r, RR, aR) +(w .. 4a w —"’—)—9— f(r R a )) v i SF ar aRI ’ I’ I (2.12) where the deformation of the real and imaginary parts is assumed to be the same. To derive AUSO, consider USO written in explicit form l7 2 ) (VSO + iWso) o.($p(r) + %$) (2.13) where p(?) is the nuclear matter density distribution and both a real and imaginary potential strength are considered. If p(?) is represented by the density function f(r, R ago) and the first order expansion is made, AU so’ so becomes AU -‘h2‘*I‘6( (3”) $151 so ' (m c) so 1 so 0 OLso r 3R X i n so (2.14) Performing the gradient on aso(r)§%£— we may write .. <1) <2) 30 AU - AU + U , where so so so 2 (l) _ ‘h “ l_a_ 3f +.-> AUso . (m c) (Vso + iwso) o‘so(lfl) r 3r 3R 0 l n so (2.15) and AU (2) = (Tim +iW ) 3f *[Efl (r) xii] so m c so so 3R 0 aSO i n so The sum AUSO(1) + AUSO(2) is called the full Thomas form of the deformation. Using the usual multipole expansion * a(?) = Em a1m Y£m(r) the matrix element of d(r) is given by x . BJR m . = YJ (r) (2.16) V23+l where BJ is the deformation parameter. The interaction matrix for the central part, including Coulomb excitation, is then just l8 BJ [V R §£_ cent (2J+1)l/2 o R 3BR 3 3f ‘ 1 RI(wV ‘ uaIwSF SEQ 3R 2 (r/R )J r = J (afic) Rso [Vso + iWSO] /2J+1 n 2 x l 3f M "—>.-> r 3;—§—— YJ (P)O A (2.18) so and so 2 (2) 8J «h . = ( ) R [V + iw ] so (2J+l)l/2 mflc so so so 3 f BRS if + + M . 1 + O o-(V(YJ (r)) X f v) (2.19) Since contains an Operator which O differentiates the distorted waves, it is not simply calculated in the "standard" codes, such as JULIE, DWMAIN or DWUCK. These codes calculate the radial form factor and distorted waves separately. The overlap integral is then numerically preformed on the product of the incoming and outgoing distorted waved and the form factor. H. Sherif has written a DWA code (SH 68) which includes the full Thomas form of the distorted spin—orbit potential. 19 The terms which include the derivatives of the distor— ted waves are included. The deformation parameters BJ and 830 are left as free parameters so that the rela— tive deformation strength may be varied. 2.6 The Microscopic Model The matrix element in equation 2.5 may be calculated using nuclear wave functions which are a super- position of the wave functions of the individual nu- cleons in the initial and final states. The interaction potential V Operates between single nucleon initial and final states. These two nucleon matrix elements, weigh- ted and summed, comprise a microscopic discription of the scattering reaction. In the zero range approximation, using an inter- action potential which ignores the L-S force, the form factor G2sj,m of equation 2.7 may be factored and cal- culated separately. In order to separate the radial and angular dependence of the interaction potential, it may be written in a multipole expansion (SA66) —> + —> __ 2 J...“ -> V(r’ Xa’ XA) ’ LSJ,u (’) VLSJ,u(r’ XA) " + TLSJ_u (r, xa) (2.20) where __ Z .L m ‘ + TLSJ_L1 — m l Y£(r) SSu_m (xa) 20 is the spin—angle tensor. By defining (PE 71) the tran- sition density 5(r - r ) LSJ — Z O i LSJ . F (r0) = /2 O (2.21) the radial form factor, GLSJ(r), becomes _ LSJ 2 GLSJ(r) - IVLS(r, r0) F (r0) r0 drO (2.22) The multipole coefficients v are the coefficients of the LS multipole expansion of the potential. One potential used here is the Kallio - Kolltveit (KK) (KA 64) interaction. This potential is written vTE = 475 exp(—2.5214 (r — 0.4)) VSE = 330 exp(-2.4021 (r - 0.4)) (2.23) for r > 0.4 and VTE = VSE = m for r 5 0.4, where TE and SE refer to the triplet and singlet parts of the interaction in total spin which have even symmetry in the spatial coordinates. Another potential which may be used has the Yukawa form 21 -ar — £_ V(r) - V Gr In comparison to the KK potential, if a 1 fermi range is used for the Yukawa potential, then the strengths of __ -> —> V(r) - V0(r) + Vl (r) 01- o 2 V0 V1 (MeV) pp —18.4 18.4 pn —54.0 —5.75 (2.24) will produce equivalent results (PE 71). The nuclear wave functions used to calculate the transition density were those of D. M. Clement and E. Baranger (CB) (CL 68). These wave functions were calcu— lated in a space of twelve single particle orbits for both protons and neutrons. The quasi-particle Hamil- tonian was diagonalized between the excited states J" formed by a superposition of two neutron quasi-particles coupled to J1' and proton particle-hole excitations coupled to J1r anui a closed core ground state. The tran- sition density is (CO 70) LSJ z L+S F r = z U.V. + — U.V. ( ) JJ' 33' ( J J' ( ) J J) -1/2 .. l + .. 2.2 TJJ. ( oJJ.) ( 5) with 2' l/2 j' .. AA can I 23., = l J'AAILSJ (-)Z 20 g 3 L S J J /T A 1/2 J un£(r) un,l,(r) (2.26) 22 where Uj and VJ are the occupation parameters (BA 60), wjj' is the amplitude from Clement and Baranger, and uni is the radial part of the nuclear wave function calcu- lated in an harmonic oscillator potential well. The FORTRAN codes FBART and NUCFAC (PE 70) were used to calculate the transition density and form factor for input to DWMAIN. Values of Wjj' and V3 used in these calculations are in Table 2.1 Table 2.1 Clement and Baranger Wave Functions and Occupation Numbers for 120Sn and l2“Sn 2+ States neutrons l208n 12”Sn I 0" Bi VJ “'ii' Vi p3/2 p3/2 -.013 .99634 I .01 .99683 pl/2 -.O23 -.02 f5/2 -.011 -.01 f7/2 .046 .04 pl/2 f5/2 -.O24 .99503 -.02 .99563 f5/2 f5/2 -.019 .99565 .02 .99612 h9/2 .139 .14 f7/2 —.001 00 g9/2 g9/2 .040 .99416 .04 .99523 d5/2 .081 .06 87/2 .037 .03 113/2 —.l44 — l3 d5/2 d5/2 .080 .97475 .06 .98314 g7/2 .045 .03 31/2 .187 12 d3/2 .132 .09 s7/2 g7/2 .155 .95506 11 .97140 d3/2 .309 21 sl/2 d3/2 .321 .89252 22 .94295 d3/2 d3/2 .299 .69423 24 .82497 hll/2 hll/2 —.602 .50763 - 74 .67294 h9/2 —.074 — 11 hll/2 f7/2 -.l70 - 20 h9/2 h9/2 —.092 .18520 - 13 .23899 h9/2 f7/2 .015 02 113/2 113/2 .058 .10566 06 .11668 f7/2 f7/2 —.040 .11569 — 04 .13197 protons 12OSn 12“Sn 1 1' “11' “’JJ' 93/2 r7/2 —.063 -.06 f7/2 h9/2 -.155 -.16 f5/2 f7/2 .005 .01 89/2 d5/2 .311 .28 89/2 g7/2 .108 .10 g9/2 113/2 -.164 -.16 3. EXPERIMENTAL PROCEDURE 3.1 Cyclotron and Experimental System Proton beams for this experiment were accelerated by the Michigan State University sector focused cyclotron (BL 66). Normally 100% Of the internal H+ beam was ex- tracted via an electrostatic deflector and magnetic chan- nel. Figure 3.1 shows the floor plan of the cyclotron experimental area and beam line used. The beam transport system (MA 67) focused the ex- tracted proton beam from the cyclotron on slits 81. After being bent through 90° by magnets M3 and M4, a second focus was formed at slits S3. Beam divergence was limited by slits S2. Typical slit openings for this experiment were .100 inches for all slits. These slit values limit the FWHM energy spread to 8 parts in 10“. Proton energies were determined from nuclear mag- netic flux meters in the central fields of M3 and M4. The energy of the analyzed beam as a function of mag— netic field strength has been calculated (SN 66). Recent measurements based on a new technique (TR 70) have allowed the calibration of the absolute energy Of the analyzed beam to better than 1 part in 1000. The analyzed beam was deflected into the target cham- ber by magnet M5, and focused at the target center. No collimating slits are used between M5 and the faraday cup in order to minimize radiation background in the experimental 24 .mpoumhonmq coppoaomo zpfimno>fiCD mumpm cowacOHz map so some HmpcoEHAOme H.m opswflm ‘( \\\\\\N A! \x \\ \ \\ \\ \\\\\ \\\\\ \mewmwwvmwM 26 area. The focused beam was typically rectangular, 1 mm high andZSmm wide. The beam was positioned by Observing the beam spot relative to fiducial marks inscribed on a one half mm thick piece of plastic scintillator viewed with a closed circuit television system. This allowed positioning of the beam spot to within 20 mils. Targets for this experiment were isotopically en— 120 riched self-supporting rolled foils of Sn 9.9 mg/cm2, 12”Sn 5.13 mg/cm2, 94.7% iso— 98.4% isotopic purity and topic purity. The isotopes were Obtained from the Iso- topes Division of Oak Ridge National Laboratory and the targets fabricated at Microfoils Inc.,Argonne, Illinois. Since target thickness and uniformity are unimportant in the reduction of the data, thickness was determined by weighing only. The targets were mounted in the existing target chamber (KO 69) which allows remote positioning of both target height and angle. Beam exiting the target chamber was collected in a 2.9 inch diameter by 59 inch long faraday cup. The beam stop was located 2 m beyond the target position. With this distance, radiation from the faraday cup reaches the gamma detector approximately midway in time between ra- diation counts from other beam bursts at the target. The beam stop was a .75 inch carbon block chosen for its low neutron production characteristics. The whole faraday cup was encased in a 22 inch diameter by 34 inch 27 long cylindrical water shield to further reduce the neu— tron flux. The beam current and integrated charge were measured with an Elcor Model A 310 B current integrator connected to the faraday cup. 3.2 Detectors Gamma rays were detected in this experiment with a Harshaw Integral Line 2 inch diameter by 3 inch long NaI(Tl) crystal coupled to a RCA 8575 photo multiplier tube. This detector has a measured energy resolution of 7.6% for the 662 KeV gamma ray from 137Cs. Bias voltage was supplied to the photo multiplier by an ORTEC 265 photo tube base. This base is designed so that the photo tube anode is maintained at ground po- tential, thus the anode signal rise time is not limited by the time constant derived from a large coupling ca— pacitance. The ORTEC 265 base allows external voltage stabili- zation of the final four photo tube dynodes. Current flow from the voltage divider resistance chain through the dynodes during pulse amplification increases the total current in the resister chain. This increases the poten- tial between the first few dynodes and the photo cathode. The effect of this is a rather strong gain increase with increasing count rates. This effect may be reduced by adding large capacitors in parallel with the resistors in 28 the voltage divider chain. Since the fraction of the ca- pacitors' charge necessary to compensate for the lost dynode electrons is small, these serve to hold the inter- dynode voltage difference more constant. Expanding on this technique,batter1escn'Zener diodes may be used to supply the voltage to the dynodes. In practice, Zener diodes and .5 pf capacitors were placed in parallel with the last four dynodes. The zener diodes were chosen such that the interdynode voltages were those supplied by the base with an overall Operating voltage of -3000 V. The operating voltage used was —l800 V. This increased the relative amplification of the last four, stablized, dy— nodes in comparison to the normal amplification at -1800 V. In this configuration, the shift of the centroid of the 1.33 MeV gamma line from 60Co was 5% (compared to 100% without stabilization) when the counting rate was varied from lOO/sec to 65,000/sec. To reduce background, the gamma detector was en- cased in a lead cylinder with 4 inch thick walls and 24 in- ches long. The shielding weight was 1/2 ton and was moved along a line centered to within 10 mils of target center by a motorized screw jack. The measured total displacement Off center is 10 mils. With this shield, the background from sources other than the target was measured to be 10% of the total counting rate when using a tin target. 29 The product of the gamma-ray detector efficiency and solid angle was determined directly in the experimental apparatus. A 60 Co radioactive source calibrated to i2% at the MSU Cyclotron Lab (KO 69) of known activity was placed at the position of the center of the target. Ob- servation of the 1.17 MeV gamma—ray from this decay pro— vided the detector efficiency-solid angle product, 5A0, for the 1.17 MeV gamma-rays. Only the photo peak in the gamma—ray spectrum is used in calculating 5A0. The efficiency-solid angle for 1.13 MeV gamma-rays was ob— tained from this with a small correction (HE 64). The efficiency-solid angle product for the gamma—ray photo peak and the face of the gamma-ray detector at 6 1/8 inches from the source was determined to 16% accuracy to be l.16x10"3 steradian for 1.17 MeV (120 l2uSn). Sn) and 1.24x10-3 steradian for 1.13 MeV ( To detect charged particles a 5 mm x 500 mm2 li— thium drifted silicon detector was used. A detector of large surface area was necessary in order to obtain a sufficiently large proton solid angle. To provide a suitable environment for operating the detector outside of the target chamber, a portable vacuum box was constructed (see Figure 3.2). Protons entered the box through a 1 mil aluminum window. Solid angle definition was Obtained with collimators placed in a hold- er in front of the particle window and external to the vacuum. The detector was housed in the box in a brass .LOp00poO oaoapmma map mcfimsoz xon 8350m> mo 3OH> OOflm m.m opswfim \ — \ NW” \ \AIIMI — .u. Ii 2. \ u / mmOJOI 65.52.4400 96 "III! 382.3 szoE /. / m/ \ \\ mOhomkwo 55:04) / kmOn. Iwaomzhowwm rooomos I ozm 232.55.? Q mm: II On» .EspuOOQm chasm cowopd HmOHQ>B m.m mmeDz 49.256 000 osmfioe Opdwfim OnN JI- I_ii§ ,— . -‘ . 59- <' ASH Ll'l — £13." 0 N «I- b 4 com a :40 ankomaw >omwzw ZOFOE 53. no. to. “IBNNVHO 83d SLNflOO 32 holder to which alcohol cooled by dry ice was pumped from an external source. The detector holder was fastened to the wall of the vacuum box with Delrin plastic mounts which provided both heat and electrical insulation for the detector and mount. After a period of rough pumping va— cuum was maintained with a cryogenic pump filled with mo- lecular sieve and kept at liquid nitrogen temperatures. A typical spectrum of protons scattered from 12OSn and detected with this system is shown in Figure 3.3. The resolution for the elastically scattered protons is 250 KeV FWHM. 3.3 Electronics Because the coincidence count rate is limited mainly by the limit on the count rate in the gamma-ray channel, the electronics were designed to extend this limit to as high a count rate as possible. In particular, dead time and pile up effects must be minimized. To reduce elec— tronic dead times, cable delays and cable delays with amplifiers were used instead Of gate and delay genera— tors On all timing and logic signals in the gamma channel. Also, 100 nsec differentiation and integration constants were used to shape the bipolar gamma pulse. A block diagram of the electronics configuration ap- pears in Figure 3.4. Fast timing signals from a timing single channel amplifier (TSCA) set on the bipolar proton amplified pulses started the time to amplitude converter 7 N01 7- DETECTOR a DYNODE No I PROTON MONITOR and PRE AMP PNOTO TUBE 00L ANODE AMP » TIME PRE AMP PICKOFF l mu 55"” TAC. “5T 05”" SHAPING or op DISC A AMP DELAY Ts“ AT SLow Afim DELAY TSCA AMP COINC. AMP GATE DELAY ENABLE , AMP PULSE LINEAR _. LINEAR LINEAR GATE GATE GATE STRETGHER STRETCIER STRETGHER UNGATED GATED GATED I I ADC CHANNEL ZERO mm m A oc an F”_ _"_'- -'_"' _""1 l LINEAR 2-0 DATA STORAGE ' | DATA STORAGE PROGRAM l ' PROGRAM I ' I l 2 7 COMPUTER l____.__________..l Figure 3.4 electronic system. Block diagram of the SHAPING AMP ,,,,, I )IIIII: SIEEEIII'E 34 (TAC). Although the TSCA cross—over timing pulses were quite pulse-height dependent, the high efficiency and simplicity in use of cross-over timing over other timing methods were judged more important than improved time resolution in this case. Delayed signals from an induc- tive pick up coupled to the anode of the gamma detector photo tube were used to stop the TAC. A typical time spectrum is shown in Figure 3.5. TAC starts were se— lected from protons scattered from states of 0 to 3 MeV excitation in the tin target. The spectrum passed through a linear gate enabled by a TSCA which selected pulses be- tween 0.7 and 1.5 MeV in the gamma-ray energy spectrum. The large peaks are due to the pulsed nature of the cy— clotron beam which has a period Of 61.5 nsec at 30 MeV. Structure within these peaks was observed to correspond to TAC starts from protons scattering from separate energy levels in the tin nucleus. This was done by Observing the time spectra of each proton state in relationship to the cyclotron rf. This structure is due to charge col— lection effects in the silicon detector integrated into the double—delay-line amplified proton pulse. These charge collection effects are seen in doped germanium and silicon detectors where the mobility of electrons and holes are different. The shape of the detector out— put has two slopes on the leading edge of the output pulse resulting from the separate carrier mobilities. This difference in mobility results in a difference in the .ESROOOQm pnwfion omHSQ pzapzo o Ex, Ep) = T(AEy > Ex, Ep). In practice, the calculation of R(Ey, AEp 2+) as described in the above example, was used to obtain R. R(AEy P.P., Ep) was also calculated to check the limits of AEp for the 2+ state. A computer program was written which allowed input of either the one dimensional coinci— dence spectra or the 2—D coincidence array in order to calculate the real coincidence gamma—ray spectra for the first and second excited states. A second, similar pro— gram calculated real coincidence proton spectra. an The photo-peak was cflWxnl distinguishable in the real coincidence gamma—ray spectrum for the first excited state, R(Ey, AEp 2*). When this was obtained, a bin of gamma-ray pulse heights was chosen to include only the photo—peak. The number of real counts in this bin was used to calculate the spin flip (see equation 3.1). In cases where the number of real coincidences for the 2+ state was small, the real coincidence spectrum containing — + the cascade gamma—rays (3 + 2 , 2+ + 0+) of the strongly (l2OSn 2.39 MeV and l2uSn 2.55 MeV) state was excited 3- heavily relied upon to define the limits of an acceptable gamma-ray bin. The photo-peak of the 2+ to 0+ member of the cascade decay was always clearly distinguishable in R(Ey, AED 3_). This peak has the same gamma-ray energy and thus pulse height as the (2+ + 0+) decay obtained by directly exciting the 2+ state. Since this peak is sensitive to all the same experimental conditions, such as gain shifts, as the peak expected in the real coinci— dence spectrum for the first excited state, the limits obtained from inspecting this peak could be directly ap- plied to the real coincidence spectrum for the directly excited 2+ state. Real counts along the Ep axis were scaled and added to those in the gamma-ray energy bin. It is presumed that these counts represent a random sampling of the counts throughout the gamma-ray energy spectrum. Thus real counts along the axis were scaled by the ratio of T 1 T {illiil’El‘t .(IEE’i’. 45 the number of real counts in the selected gamma-ray energy bin to the total number of real counts. Addition of the real events along the proton axis typically lead to an increase in the total number of real events of 5% and never more than 15%. The statistical standard error, 68(6), associated with 8(9) (equation 3.1) is 2 _ §1 l 2 + 2 [63(6)] - 5 m] [T(2 ) + K U + U2(5K)2] / N , (3.5) where the pulse height summations over AEp and AEY in T and U are implicit and the error in N is small and omitted from this formula. The formula for 8(a) in terms of R/N when detectors subtend finite solid angles is not as simple as formula 3.1. The radiation pattern for gamma—rays resulting from a spin-flip (Am = :1) transition is peaked along the line perpendicular to the scattering plane. Also the radiation patterns for the non-spin—flip (Am = O, :2) transitions are not zero near the perpendicular (SH 70). Thus, to calculate 8(6) for a finite gamma-ray detector and a point proton detector, one must take the weighted average of . the spin-flip gamma—ray radiation pattern over the detec— tor solid angle and subtract the contribution of real co- incidences from non-spin-flip transitions. The number of real coincidences in a finite gamma—ray detector which can result from non-spin—flip transitions is a function 46 of the relative populations of the m = O and m = :2 (non- spin-flip) sublevels of the excited 2+ state. Each has a separate distribution in de-excitation. The position of a particle detector with infini— tesimal aperture defines the scattering plane. An aper- ture of finite size will define an envelope of scatter- ing planes. Each scattering plane in the envelope is weighted by the fraction of the total accepted particle flux contributed from that plane. The size of the enve- lope depends on the scattering angle, becoming larger as the scattering angle changes from 90° in the laboratory. In this experiment 8(9) was calculated with a formula (HI 70) which is a function of scattering angle, the ratio of the m = +2 to the m = O substate populations, the half angle acceptances of the proton and gamma detectors, the geometry of the proton detector aperture and the depen— dence of the gamma-ray detector efficiency on the angle of incidence of the gamma—ray. The derivation of this formula assumes that the gamma—ray detector is circular and that the m = +2 and m = -2 sublevels are populated equally. The formula is: 8(9) 2[l.6(l+2q) R/N — 3AuB - 6A B + 2.25A5B l 2 2 + 18A6BU + 6A3B5 + q(.75A5B3 — 2Al + 6A6BU + 2A3B5)]/[2Al — 9AUB — 18A2B 3 1 + 7.5ASB3 + 6OA6Bu + 2OA3B5 + q(7.5A5B - 6AuBl — 12A2B2 + 60A6Bu + 2OA3BS)] 2 3 (3.6) 47 where x x 2 A = f Y sin x p(x)dx B I p f(x)x dx 1 O l O . 2 2 811’] 6 +X X 2 Xp f(x)sin26dx A = f Y cos x sin p(x)dx B I 2 O 2 O 2 2 sin 6 +x X u X Ll A3 = f0Y cos x sin p(x)dx B3 [Op f(x)x dx (sin29+x2)2 x x 2 2 A = f Y sin3xp(x)dx B j p f(x)x sin edx M O A O . 2 2 2 (Sin 9+x ) X 5 Xp f(x)sinu6dx A5 = [CY sin xp(x)dx B5 - f0 2 2 2 (sin ed+x ) x 2 3 A6 = oncos xsin xp(x)dx and P(x) = the angular dependent gamma—ray detector efficiency th xi = the half angle acceptance of the i detector f(x)dx = a weighting function defined by the shape of the proton detector aperture which gives the fraction of protons between angles x and x+dx q = a2/aO is the ratio of the population of the m=+2 nuclear substate to the m=O substate. Figures 3.8 and 3.9 show the spin flip angular distri— 120 12“Sn when calculated with formula butions for Sn and 3.1 which assumes that both detector apertures are infini- tesimal. Figures 3.10 and 3.11 show the same angular distributions when calculated with formula 3.6. In the later calculations SPIN FLIP PROBABILITY I I I I I I I I 0 . Sn (p,p) Ex = U? MeV 8(0) WITH NO SOLID ANGLE CORRECTION (POINT DETECTORS) l L l l L 1 J1 l 60 90 PROTON SCATTERING ANGLE (DEGREES) Figure 3.8 1208n spin-flip data with- out detector solid angle corrections. SPIN FLIP PROBABILITY I I I I T I I I mSn (p.p) Ex = I.I3 Mev 3(9) WITH NO SOLID ANGLE CORRECTION (POINT DETECTORS) I, 1 L l l L l l 1 60 9O PROTON SCATTERING ANGLE (DEGREES) Figure 3.9 l2”Sn spin—flip data with— out detector solid angle corrections. |80 50 P(x) = (edQ)y /‘de = constant, f(X) =u/Xp'fl /l- (X/X )2 p and a2 / a0 = q = 1. The half angular acceptances are xp = .07 rad. and Xy = .134 rad. The errors shown in Figures 3.8 and 3.9 are only statistical. In Figures 3.10 and 3.11, the errors include the RMS sum of the statistical error and an estimate of the error generated in Choosing q = 1.0. In neither case was the 16% normalization error resulting from the uncertainty in the gamma-ray detector efficiency included. Since the actual substate populations are unknown and can be measured only with substantial effort, the values of 8(6) calculated at each angle with q = O and q = w represent the limits of possible values of 8(6). The values of these limits with purely statistical errors are shown in Figures 3.12 and 3.13. Plus or minus one third the difference between these limiting values was used as an estimate of the uncertainty generated by the arbitrary choice of q. This contribution doubled the error on some forward angle data where the statistical error is small, but increased the error in the back angle data by about 10% of the statistical error. SPIN FLIP NPROBABILITY I I I I I 7 I I I 20 , Sn (p,p) Ex = H? MeV 8(9) WITH AVERAGE (q = I) A SOLID ANGLE CORRECTION 1.11)) 1 1 l l L l l SO 90 I80 PROTON SCATTERING ANGLE (DEGREES) Figure 3.10 1208n spin-flip data with an average solid angle correction (q = l). SPIN FLIP PROBABILITY I I j I I I T '24Sn (p.p') Ex = US MeV SIG) WITH AVERAGE (q=l) SOLID ANGLE CORRECTION I I l J l 1 l L L 60 90 PROTON SCATTERING ANGLE (DEGREES) Figure 3.11 l2“8n spin—flip data with an average solid angle correction (q=l). SPIN FLIP PROBABILITY I I I j I I I j IzoSn (P.P') Ex = H? MeV MAXIMUM (q = 0) AND MINIMUM (q = (D) VALUES OF S(9) FOR FINITE GAMMA RAY AND PROTON DETECTORS :1 I I | :1) 1 l L L 1 ** 1 L 60 90 ISO PROTON SCATTERING ANGLE (DEGREES) Figure 3.12 l208n spin—flip data with maximum (q = w) and minimum (q = 0) solid angle corrections. SPIN FLIP PROBABILITY to ‘I I I I I I '24Sn (p.p') Ex = I.I3 MeV MAXIMUM (q= O), MINIMUM (q =ao) VALUES OF 8(9) FOR F INITE GAMMA RAY AND PROTON DETECTORS l J J L 1 60 SC PROTON SCATTERING ANGLE (DEGREES) Figure 3.13 12“ maximum (q = 00) and minimum (q solid angle corrections. Spin-flip data with 0) 55 Calculating 8(6) with Formula 3.6 resulted in lower values than those obtained by assuming that the detectors had infinitesimal apertures. This is especially true for the forward angle l2“Sn data where the Choice of any value of q greater than zero results in values of 8(6) (see Figure 3.13) which are negative outside the experimental error. The data on 1208n and lzuSn are quite similar. The main difference between the two isotopes is seen at angles forward of 75° where the 120 tantly higher than that of 124Sn. 8n data is consis- 4. DATA ANALYSIS A.l Inelastic Scattering Data The spin flip data treated in this section repre- sents many lengthy periods of data collection on the cyclotron. Data taken at forward angles represent the shortest data collection time, requiring about four hours per angle. At backward angles as long as twenty hours of data collection time was spent on one point. Because of the length of time invested in obtaining a data point at some angle and because of the regular nature of the data at forward angles, the angular distribution was taken at 15° intervals. The data sets considered in this section have been corrected for experimental solid angle effects as described in the experimental section of this work. Elastic and inelastic cross—section data was ob— tained along with the spin flip data at each angle in the spin flip angular distribution. The large solid angle acceptance of the proton detector used in this ex- periment averages the cross—section measurement over its angular aperture and degrades the usefulness of the cross— section data. I Both inelastic cross-section and inelastic asym— metry data are available in the literature for protons at 120 30 MeV for 8n (KA 70). These data are included in this analysis of proton scattering. 56 57 4.2 Elastic Scattering Data and Optical Model Parameters The basic set of scattering data at 30 MeV, including elastic cross-sections, reaction cross-sections, and elastic polarizations on targets from C to Pb was taken at the Rutherford High Energy Lab (RHEL) (CR 6A, RI 6A, TU 6A). The polarization measurements were repeated with thinner targets so that the average proton beam energy (GR 70) compares more closely with that for the cross— section data. These data wereagadrlretaken (KA 71) at RHEL in an experiment which included cross-section and asymmetry measurements on the strong inelastic states of some of the same nuclei. Optical model searches were done by the RHEL group (BA 64) and later by G. R. Satchler (8A 67) and G. W. Greenlees (GR 66, CR 70) in separate attempts to extract optical model parameters from the data in a consistent manner. None of the studies reported at 30 MeV proton energy include the 12“Sn nucleus. Elastic cross section and polarization data have been collected and searches have been done over the till isotopic sequence, including 1208n l2“Sn at 39.6 MeV (BO 68). The 39.6 MeV data, the and RHEL data and data from other sources at these, and at other energies were combined. A search was conducted on all the collected data testing various analytic expres- sions for the Optical model parameters by E. D. Becchetti 58 and G. W. Greenlees (BGOM) (BE 69). The result of this search was a set of best fit formulas from which Optical model parameter values may be calculated for any nucleus in the range A = “0 to 208 and for energies at least up to “0 MeV. The results of this search are expected to produce the most consistent relationship between the Optical 120 model parameters used in the calculations for 8n and l2“8n. Other sets of parameters have been tried and calculations with one set (GR 70) are also presented here. The formulas for the BGOM parameters are vR = 5“. - 0.323 + o.uz / Al/3 + 2“.0(N - Z) / A rR = 1.17 ar = .75 WV = .22E - 2.7 3 0 wSF = 11.8 — 0.25E + 12.0(N - Z) / A '= = rI rI 1.32 I: = —. aI aI .51 + .7(N Z) / A V = 6.2 so r = 1.01 so aso = .75 The values obtained from these formulas for 1208n at 30 MeV are compared with the values obtained from other searches in Table “.1. In general, the values of a given parameter do not differ greatly from one another. The main deviation between separate parameter sets is seen in Asoomv mm. H04 m6 mam. mmé No.0 m.m mm. waé mo.mm mm mm enema ow :ma H mm m smm aom H mm ma 0 o om :wa H o a: as mSvOm> Amva AEVHL A>®Svmm3 A>®Zv>3 Amvmw Amvmm A>m2vm> CmONH mmmBmEoo< ; : 650/3 = I —— _ b +**, 350/3 3 2 +++ 1 II‘*§ I I ‘ '1, 1 Ex" I 2 C I ‘ I- I ‘ : I. ** I I I a t I . . I + r. :3: x a 4‘ x I x. I I I I : x x 1 I“ 5:. L L —I o L 66 1 I20 I L l Proton Scattering Angle (Degrees) Figure “.“ tions of the Collective model calcula— 1208n cross section inclu- ding the deformed spin orbit with BGOM. 66 seen in Figure “.5. The main feature of interest is that while the strength of the spin orbit well of GR 70 #3 is less than that of the BGOM, the value predicted for the backward peak is higher than that predicted by the BGOM. The relationship between the depth of the spin orbit well and the magnitude of the predicted back angle peak is not a straightforward one. The effects of varying the optical model parameter set on the calculated values of spin flip are interesting. The collective model without a distorted spin orbit term correctly predicts the backward peak. This form of the interaction potential allows no spin flip in the nuclear interaction. Thus, the back angle peak must be the re— sult of spin flip in the elastic channels. It is known that in the absence of spin orbit coupling in the elastic channels that the backward angle peak is not predicted (KO 69). However, the magnitude of the backward angle peak does not depend on the depth of the spin orbit well alone. It also depends on the strength and shape of the other wells and the strength of the full Thomas spin orbit term. Calculations of the asymmetry and cross section with the GR 70 #3 parameters are seen in Figures “.6 and “.7, respectively. For either the asymmetry or cross section, agreement with the data is not affected much in using different optical model parameters. The value of B(.l27) is 5% lower than B from the BGOM calcu- lations. SPIN FLIP 120 tions of the Sn spin—flip with opti- cal model parameters BG 70 #3. I I— f I "T'-—" IT'T‘“ — 'T'"' .._.. T—-- ' 1 I20 . . 8n 2+ SpIn Fle Collective model (GR 7046) .4 40/3 : 0 XXX 330/8:- | —— L Bso/B = 2 +++ .2“ 0‘ "5 it “t“ + “In!” .1; i 1 60 L I20 I 4 I80 Proton ScatterIng Angle (Degrees) Figure “.5 Collective model calcula— l I I I IZO Sn Asymmetry Collective model (GR 706) '6” xxx BSD/3:0 I, x , I —— Ego/B = I + .* +++ [350/3 -.-. 2 + + ,+ + + I :2»- + + I, x . g ++++++ + + x + 5‘ 0144+ + I! < + x " Xx x x l .. 2 >- I xx xx + L L 0 Go . I20 1 I Proton Scattering Angle (Degrees) Figure “.6 Collective model calcula— tions of the 120Sn asymmetry with op- tical model parameters BG 70 #3. IZO . IOL Sn Cross sectIon , I Collective model (GR 706. ; xxx figoAB= O : ‘ — A./P= . ‘ 4~+++ ‘ .l 1m," +++ +++ 350/3 a 2 J I E I r n 1 Q l: I I j 9 - -I g I x “ s ’ , I 1 \b I *3} + " U I +++ 3:" ++ + x J Xxx I + x x II .I T {If .2) O l L 610 L l |éo l l | o Proton Scattering Angle (Degrees) Figure “.7 Collective model calcula- tions of the 120Sn cross section with optical model parameters BG 70 #3. 70 M.“ Microscopic Model Calculations Microscopic model DWA calculations have been done for 120 124 the spin flip probability on Sn and Sn using the KK force and the Clement and Baranger wave functions. The optical model parameters used here are the same as in the collective model case, i.e., the BGOM and alternately, GB 70 #3 from Table u.1. Comparisons are also made to the published cross section and asymmetry data on 120Sn. The results of the spin flip calculations on 120Sn and l2“Sn are presented in Figures ”.8 and 4.9. Exchange is explicitly included in the calculations for both nu— clei using the Petrovitch approximation (PE 71). These calculations were repeated for 120Sn including exchange exactly using the code DWBA 70 (SC 69) which is written in the helicity formalism (BA 68). Because of restric- tions in the input to DWBA 70, the KK force was not used. Rather, a force of Yukawa form was required. The range used was 1 fermi and the strengths were chosen to produce the same results for the direct calculation as the KK force (PE 71). So, while the comparison is not exact, agreement between the calculations would indi— cate that the exchange approximation is not in serious error. This is the case for these calculations (Figure u.8). The difference between the exact and approximate calculations is minor. SPIN FLIP I Y Y T Y I I Y I20 . . Sn Spin Flip EXCHANGE APPROX. u xxx BGOM ' . +++ GR 70#3 +1. E XACT EXCHANGE — BGOM 1 1 l l l l so {20 l80 Proton Scattering Angle (Degrees) 120 Figure “.8 Microscopic model Sn spin-flip calculations. SPIN FLIP Proton Scattering Angle (Degrees) Figure “.9 Microscopic model spin-flip calculations. 12“ Sn l24 . . Sn Spt n Flip EXCHANGE APPROX. x .4 r BGOM xxx it x 4 Jo xxxxxxxxx‘x K“ix‘xx xx““xxxxf‘ ) ‘X 0 L. It" I I 1 I .t o L 1 6L0 i L '20 I80 73 The GE wave functions with the KK force predict a spin flip which is in general agreement with the data. 120 The backward peak is somewhat too high for Sn, but over all the fits are good. A microscopic discription of the spin flip inter— action might by expected to reproduce the details of isotopic effects in the data. The calculations for 1243n 120 is not depressed over that for Sn at forward angles. An explanation of the lower values of the data for l"2L4Sn does not result from these calculations. The microscopic prediction of the 120Sn asymmetry is presented in Figure “.10. The most striking feature of these calculations is how poor the fits are to the data in comparison with the collective model fits. The values are too low throughout the whole angular range of the fit. Only the phase of the oscillations continues to agree with the data. There are two factors included in the collective model case but not in these calculations which might affect this. First, the KK force is central. No spin orbit force is included in the KK force. Second, the KK force is real. The force represented by the col— lective model is complex and includes a spin orbit term, the strength of which was varied. The cross section calculation is presented in Figure “.11. The normalization of the cross section is absolute, containing no effective charge parameterization. With this in mind, the agreement of the cross section calculation Asymmetry EXCHANGE APPROX . .6 xxx BGOM [ I ‘ +*+ (NR7TN’3 EXACT EXCHANGE — BGOM I .4 l I a .2». [1 [xx + 4) o l l . l , i y -.2 ‘¥Ixx. + ‘} 0 ‘x + ” 4 T I r r T I I T 'ZOSn Asymmetry 60 l20 Proton Scattering Angle (Degrees) Figure “.10 Microscopic modellgOSn asymmetry calculations. da/dn (mb/Sir) TITFT—TI T V T T ‘1' :20 . Sn Cross section EXCHANGE APPROX. ,“,1 xxx BGOM W3 +++ GR 7O 3 ‘1 EXACT EXCHANGE " — BGOM II f "3000mm;I x+I L J L #1 1 LLLLLL L {(11111 so _ ‘ l Proton Scattering Angl Figure “.11 Microscopic model cross section calculations. 120 éo * * e (Degrees) Sn 76 with the data is very good. The cross section is too low at the first maximum, but the shape is in general agree— ment with the data. The CB wave functions appear to de- scribe the 2+ state very well. The exchange approximation is also successful here. The approximate and exact calcu- lations are in close agreement. Included in Figures “.8, “.10, andligylare calcu- lations with the CB wave functions and KK force with ap— proximate exchange using the optical model parameter set GR 7O #3. As in the collective model case, varying the optical model parameters does not result in large changes in the calculations. The parameter set with the lower VSO strength again predicts a slightly higher value for the backward angle spin flip peak. To investigate the degraded fit obtained with the microscopic force and wave functions to the asymmetry data, the exact exchange calculation was repeated with a microscopic L-S force included in the interaction. The radial form of the L-S force used is a superposition of two Yukawa forces. The volume integrals of V(r) r2 and V(r) r” have been equated to the volume integrals of the Gaussian force of D.Gogny (G0 70) and the strengths and ranges of the Yukawa form determined. The potentials with strengths in MeV and ranges in fermis obtained through this method are (AU 71) 'a: Asymmetry i3 9 l20 Sn Asymmetry EXACT EXCHANGE xxx TWO BODY SPIN ORBIT I 1 ~ -— CENTRAL KK Xx‘x‘xxx 1 ‘xxx i J L 1 640 .L '20 1 J '80 Proton Scattering Angle (Degrees) Figure “.12 Microscopic model 120Sn asymmetry calculations with a two body spin orbit force. SPIN FLIP I20 - . Sn Spin Flip 4 EXACT EXCHANGE " ° xxx TWO BODY SPIN ORBIT '——- CENTRAL KK .2r *“I“x“¥x x' I o 60 iéo Proton Scattering Angle (Degrees) Figure “.13 Microscopic model 120Sn spin-flip calculations with a two body spin orbit force. I20 . .0_ Sn Cross section 1 E EXACT EXCHANGE j i 1 xxx rwo BODY SPIN ORBITA * — CENTRAL KK 4 r- 1 -i int“ 1 I I I A I —4 ,2: I:— I : 42 : I j 9 ~ I E E 1 i <3 b 111 - E i- I I I I I 1 4 I I I I: J :- I ? o l I 60 J l iéo l80 Proton Scattering Angle (Degrees) Figure “.1“ Microsc0pic modell2OSn cross section calculations with a two body spin orbit force. 80 < I pp - —57“ exp(.329 r) / .329 r <: ll pn ~28? exp(.329 r) / .329 r + 218 exp(.238 r) / .238 r The calculated cross section, asymmetry and spin £114) with the KK equivalent central force and spin orbit force are presented in Figures “.12, “.13 and “.1“. In— cluding the spin- orbit force does not greatly affect any of the results. w. G. Love has found the L'S force to be important in fitting inelastic scattering to excited states with high spin (6+, 8+) in 90Zr at 60 MeV (L0 71). A more complete investigation (LO 71A) showed that for the states of lower spin (2+, “+), the spin orbit contribu— tion to the cross section is not very strong. J. Raynal (RA 68), using a somewhat stronger spin orbit force (J24 for Raynal is about twice J)4 for Love), found improved agreement in the forward angles of the asymmetry of the lowest 2+ state in 90Zr at 20.3 MeV. This is not the case for 120Sn when the Gogny L-S force is used. There are two methods of estimating the imaginary part of the microscopic interaction. A simple-minded approach is to take the collective model imaginary part normalized to the microscopic calculations. A second possible prescription for calculating the microscopic imaginary part is the "frivolous model" suggested by G. R. Satchler (SA 71). The preceding microscopic calculations, including approximate exchange with the CB wave functions 81 and the KK force using the BGOM, have been repeated with each of these imaginary parts. The asymmetry calcula— tions may be seen in Figure “.15. Also, calculations done with a real collective model form factor including deformed spin orbit are seen. It is evident that a complex form factor with either imaginary part gives substantial improvement to the asym- metry over the real form factor calculation alone. Also, one sees from the real collective form factor with de— formed spin orbit calculation, that the imaginary term com— plements the effects of the deformed spin orbit. The i deformed spin orbit improves the agreement with the data at forward. angles without a corresponding effect over the rest of the angular range. The imaginary part has least effect at forward angles, producing better agree- ment over the range of middle and backward angles. The fits to the cross section are presented in Fig- ure “.16. Addition of an imaginary part produces a general improvement to the fit. The first maximum is in better agreement, while the rest of the angular range is not changed much. The spin flip calculations are seen in Figure “.17. While addition of the collective imaginary part did not affect the values of the calculations, the Satchler imaginary part grossly over predicts the value of the backward angle peak. imu3"frivolous model" clearly produces much worse agreement in this case. It is note~ worthy that while the calculations for spin flip seemed iv Asymmetry C) _ +++ REAL COLLECTIVE law/Bu I Y r F T T 'ZOSn Asymmetry REAL MICROSCOPIC XXX REAL MICROSCOPIC + COL IM REAL MICROSCOPIC 4- SA IM «PM *+¢+, . 1x \ + I . . I [I ‘ , l r“ I I x o x 7’3 \{ x I ” I. x 't " ‘ x I “4' I ,‘ f"?x~\ .\ I; x ( " / .\ .\ L 60 IZO Proton Scattering Angle (Degrees) Figure “.15 120Sn asymmetry micro— scopic model calculations including complex coupling. (80 da/da (mb/str) izo . IO Sn Cross section 4 E' xxx REAL MICROSCOPIC 3 » x +++ REAL COLLECTIVE A C —REAL MICROSCOPIC+COLIMJ . " ."i; . - —- REAL MICROSCOPIC+SA IM 1 {pi .3 * \ .I i 1‘. I I I E“ “*:‘:x‘:x; E P ‘+.,:xI 4 *- ;x ‘I t ".2. vi? I ‘ .. O“o .gI .4 . 5.. {I \ /\ 3 OK“ I - “ ’1 s ‘ ‘Rfig \ . I .I :" Ix ¥-IJ I 1 ” 1 O 1 l 1 1 L 1 1 J '80 60 IZO Proton Scattering Angle (Degrees) Figure “.16 120Sn cross section microscopic model calculations inclu- ding complex coupling. SPIN FLIP in. T fi r T I f T fir 'ZOSn Spin Flip xxx REAL MICROSCOPIC /\ +++ REAL COLLECTIVE BSD/Bu i \ — REAL MICROSCOPIC+ coL IM f --- REAL MICROSCOPIC+ SA IM ’ L so Iéo Proton Scattering Angle (Degrees) Figure “.17 120Sn spin—flip micro— scopic model calculations including complex coupling. I80 85 fairly insensitive to other changes, the Satchler imagi— nary part produced a pronounced effect. Calculations studying these complex form factors on other states and a description of the Satchler formulas are found in the Appendix. 5. SUMMARY Spin-flip probabilities for the excitation of the first 2+ states in 120Sn and l2“Sn have been measured for inelastic proton scattering at 30 MeV. The spin- flip data for both isotopes are quite similar. Both show the peak at back angles which is characteristic in medium energy spin—flip data taken on lighter nuclei. The tin cross section, asymmetry (KA 70) and spin—flip data have been analyzed with both macroscopic and micro— scopic DWA models. For the collective model, fits to the cross section are reasonably good. Use of a deformed spin or— bit term is important, but no more so than the imaginary part of the collective form factor, for the asymmetry data. Deforming the spin orbit well has little effect on either the spin-flip or cross section calculations. The fit to the 120Sn spin-flip data is quite good over the whole angular range. The fit to l2“Sn spin-flip is good for the backward angle peak but the low forward angle values are not predicted. Little structure is predicted or seen in the forward angle spin—flip data for either nucleus. For the microscopic model, the shape and magnitude of the cross section and spin—flip predictions are quite reasonable. The predicted spin—flip has a higher value at the backward angle peak than for the collective case. However, it still shows agreement with the data. 86 87 Isotopic differences were not evident in comparing spin— flip predictions for 1208n and l2”Sn. Use of an ap— proximate exchange term did not result in serious error for either cross section, spin—flip or asymmetry calcu— lations. Including a realistic two body spin-orbit inter- action potential did not significantly affect the cal- culations for cross section, spin-flip or asymmetry. The asymmetry was poorly fit with a real KK inter- action. The addition of an imaginary term to the form factor greatly improves the asymmetry prediction and shows some improvement in the fit to the cross section. Of the two imaginary terms used, calculations with the collective imaginary term fit the spin-flip data much better than calculations with the microscopic term. Cal— culations of the cross section and asymmetry for the lowest lying states of 58Ni and 208Pb were compared to data at 30 MeV proton energy. Addition of an imaginary term always improved the fit to the asymmetry. Both cross section fits were improved by the addition of the col— lective imaginary term. Only the 58Ni cross section fit was improved by using the micrOSCOpic imaginary term. 6. APPENDIX A number of authors have pointed out that when the collective model is applied to inelastic proton—nucleus scattering it is important to deform the imaginary and spin-orbit wells in addition to deforming the real well (SA 70). G. R. Satchler recently proposed a semi- phenomenological model for the imaginary form factor in microscopic (p,p') calculations (SA 71). Also, it is now possible to include a two body spin orbit term in micro— scopic calculations (SC 69, L0 71A). One such calculation for 120 Sn is described in Chapter A. In view of this it is useful to study the effects of complex coupling on cross sections and asymmetries and to compare the results with those of similar calculations which include a spin orbit term. Calculations were performed for the lowest lying ex— cited states in 58Ni, 120Sn, and 208Pb using in each case a microscopic real form factor and each of two models for the imaginary form factor. Spin orbit contributions were calculated with the collective model. The calculations were done in DWA for 30 MeV incident protons using the BGOM parameters. Exchange effects were included explicitly (PE 69). Angular distribution and asymmetry data are from (KA 70), the spin—flip data from this work. 88 89 The real part of the microscopic form factor (RFF) in DWA for a normal parity transition is FJOJ Re (r) = fVJO(r,rO)gJ(ro)r§drO (6.1) where VJO is the Jth multipole of the projectile—target interaction, which in these calculations was chosen to be the long range part of the Kallio—Kolltveit potential. The function gJ is the transition density. It was assumed that the gJ's had the same form as charge transi— tion densities used to calculate inelastic electron scat- tering form factors. These may be inferred from experi— mental data (CU 69), BA 67, HI 70, DU 67). gJ(r) = (A/Z) “CHARGE(P) (6.2) Transition densities also can be obtained from theo— retical wave functions. A calculation of this sort was 120 described for the 2+ state in Sn in Chapter U. Satchler's microscopic imaginary form factor (IN) is W(r)g (r) JOJ J Im FSA (r) = - ——3T?)—_— (6.3) where W(r) is the imaginary part of the optical potential and p(r) is the ground state matter density with the forms _ d W(r) - (WV—MaIWSa;)f(r,RI,aI) and p(r> = <1+wr with f(r,X.y) = [1+exp((1"-X)/y)]_l 9O _ 1/3 and RI — rIA . state densities (CU 69, BE 67, HA 57) are shown in Table Values of the parameters for the ground 6.1. Values of the parameters for the optical potentials were taken from the BGOM. The collective model IM is B JOJ _ J I g_ Im FCOL(r) ‘ 2J+l dr W(r) where the BJ's were obtained by normalizing the collective cross sections to those calculated using gJ's given by (H.2). The microscopic real and imaginary form factors and 58 the collective model IM for the lowest 2+ states in Ni 120 and Sn are shown in Figure 6.1; those for the 3' state in 208Pb are similar. It is instructive to examine first the effects of de— formed imaginary and spin orbit wells on the asymmetry in a purely collective calculation. Such a calculation for the first 2+ state in 120 Sn is shown in Figure 6.2. Both complex coupling and a spin orbit term are necessary to produce good agreement with the data. The spin orbit contribution is especially important in the forward di— rection. Inclusion of the IM produces a much improved fit at intermediate and back angles, an effect which is not accounted for by deformation of the real and spin orbit wells alone, even if the spin orbit strength is increased. These same observations hold for the other states studied. TABLE 6.1 Ground State Charge Density Parameters c z w 58Ni u.25 .566 o l2OSn 5.32 .575 o 208Pb 6.uo .5u2 .1u .mLOpomm Egoe mgmcflmmEfl pcm Hmmm H.@ mgswflm 5.92%.“? 6.5:“.me s: 400 ... 2. 400 III <5 4m .. l. 2. «m It aqua ... . 0.0. In 45m + N .58. macho/E Zach. '203n(p,rs) 'ZOSn‘(l.l7MeV) xxx COMPLEX 1950/3 =0 -6* — COMPLEX 350/3 =l --- REAL 1380/3 =0 +++ REAL fiso/fi : | .4; .. r '2’ 1 I 3'3 , . e . \ . E o I x . ill \ E" o ‘9 N. ‘ o o \ x I.1\ v. x ”g x ’9': If \\* o \\ i I a \ "x x/ \1 * x I j -'2 x VAR \ _// \{o If \\ \ ’ I \ " Vx ,. J O 1 1 6L0 L ; |éo L l80 Proton Scotter’ng Angle (Degrees) Figure 6.2 120Sn collective model asymmetry calculations. 9A Calculations of cross section, asymmetry, and spin— flip for the first 2+ state in 120 Sn are shown in Figure 6.3. The set on the left was done with a transition den— sity extracted from the quasi-particle wave functions of Clement and Baranger. The two real form factors are very similar to each other and to the real collective form factor. Includingeitherjnqimproves the fit to the angular distribution in the forward direction and provides a def— inite improvement in the asymmetry prediction. However, the spin-flip calculations performed with the two IM's differ markedly, the collective IM having little effect, the microscopic IM overestimating the spin-flip at back angles. Since the main features of spin—flip are believed to be determined by the optical potential (SA 70) it appears that in this case the microscopic IM effects un- desirable interference among various of the distorted waves. 58 Calculations for the lowest 2+ state in Ni and 3‘ 208Pb are shown in Figure 6.4. The effects of 58 state in complex coupling and the spin orbit term on the Ni asymmetry and cross section and the 208Pb asymmetry are similar to those for the 2+ state in 120sh. In the 58Ni case including either IM produces little change in the spin—flip prediction. While the collective IM improves 208Pb the predicted angular distribution for , the micro- scopic IM overestimates the magnitude at all angles. The I20Sn (p.p')(l.l7 MeV) 29 da’ldn (mb/slr) f x REAL FORM FACTOR (0.0') ' x REAL FORM FACTOR (ca) *.RFF.SAIM /, ,. RFF.5AIM ---RFF o COL IM A / --— RFF . COL IM —COL flto ' 3 I II J l . >_ 9.4" I I m 1‘ ‘0 lo F l . l, l .. . } l t I l ' 0:. I ‘ ‘rll ‘ .‘l I [if 11' ‘ l l o .\ l g l I‘ {.2 O J, 'lr’ J“. l - l \ I We Pox 'l 3i l ‘ \’ "II o "I‘L‘ \II \I/ a4 4 ‘ r t A s» i O F .A ' \ SPIN FLIP / Lama-H'W"M§ |20 Proton Scotter'ng Angle (Degrees) Figure 6.3 l208n microscopic calcula- tions including complex coupling. , . l H3. 1 l. d a-ldn (mb/str) ASYMMETRY 3 J 2°“Pb(p.p')<2.ea MeV) Tvrvvrv / r , v r x EN_HfiMHmKRh#) . RFF .SA IM ---RFF.COLIM Figure 6.4 Proton Scotta’ng We (Downs) 58 208 Pb and Ni microscopic calculations including complex coupling. 97 microscopic IM improves the shape of the fit only at forward angles. The shape at back angles is degraded. It is evident that complex coupling and a spin orbit term are important for accurate prediction of (p,p') asymmetries. 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