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An optical model study of the small angle elastic scattering of neutrons from lead, bismuth, and uranium… Forrester, Glen Campbell 1970

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AN OPTICAL MODEL STUDY OF THE SMALL ANGLE ELASTIC SCATTERING OF NEUTRONS FROM LEAD, BISMUTH, AND URANIUM AT h.2 MEV. *y C-LEN CAMPBELL FORRESTER B.Sc., University of British Columbia, I969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics We accept this thesis as conforming to the required standard The University of British Columbia December, 1970 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date Drcr/r)^g/f 3 5 1970 ABSTRACT AN OPTICAL MODEL STUDY OF THE SMALL ANGLE ELASTIC SCATTERING OF NEUTRONS FROM LEAD, BISMUTH, AND URANIUM AT k.2 MeV. An Optical Model study has "been made for the elastic scattering of k,2 MeV. neutrons from three heavy nuclei, Lead, Bismuth and Uranium. The Optical Model used included a real, imaginary, and spin-orbit potential well. Its parameters were determined from a least square f i t t i n g to the experimental scattering data available in the literature for angles from 15 to 180 degrees. The model i s then used to estimate the nuclear scattering to be expected at small scattering angles. It is shown that Schwinger scattering and incident neutron polarization significantly alter the shape of the elastic cross section at angles less than 10 degrees. At 5 degrees this change is about 3$» Additional effects due to Electric Polarizability scattering are shown to be very small and not sufficient for a determination of the polarizability of the neutron from the elastic scattering data at this energy. (ii) TABLE OF CONTENTS Page CHAPTER 1 - IHTRODUCTIOH 1 A. The History of the Optical Model 2 B. Optical Model Scattering Theory 7 Part 1 - Simple Theory 7 Part 2 - Extended Theory 11 Part 3 - Complete Theory 13 C. Form of the Optical Model Potentials lh D. Additional Scattering l 6 Part 1 - Schwinger Scattering l 6 Part 2 - Electric Polarizability Scattering . l 8 CHAPTER 2 - THE FITTING PROCEDURE 19 A. Introduction * 19 B. Lead - Pb(208) 21 C. Bismuth - Bi (209) 23 D. Uranium - U(238) 2k CHAPTER 3 - CORRECTIONS TO THE OPTICAL MODEL 26 A. Incident Polarization Effects 26 B. Sch-winger Scattering 27 C. Electric Polarizability Scattering 28 D. Final Corrected Optical Model Prediction . . . 29 CHAPTER k - DISCUSSION AND CONCLUSIONS 31 CHAPTER 5 - REFERENCES 33 ( i i i ) LIST OF FIGURES To Follow Page FIGURE 1 - Differential Elastic Cross Sections for J Lead - Fb(208) (a) Large Angles 21 (b) Small Angles 21 FIGURE 2 - Differential Elastic Cross Sections for Bismuth - Bi(209) (a) Large Angles 23 (b) Small Angles 23 FIGURE 3 - Differential Elastic Cross Sections for Uranium - U(?38) (a) Large Angles 25 (b) Small Angles 25 FIGURE h - Optical Model Prediction of Relative Differential Elastic Cross Sections for Lead,Bismuth,, .' and Uranium 25 FIGURE 5 - Incident Polarization Effects for Scattering From Lead (a) Large Angles . . . . . . . . 26 (b) Small Angles 26 FIGURE 6 - Sclwinger Scattering Effects for Lead - Pb(20B) . . 27 FIGURE 7 - Electric Polarizability Effects for Scattering From Lead 28 FIGURE 8 - Final Optical Model Predictions for the Differential Elastic Cross Sections (a) For Scattering from Lead 29 (b) For Scattering from Bismuth 29 (c) For Scattering from Uranium 29 FIGURE 9 - Final Optical Model Prediction for the Relative Differential Cross Section (a) For Lead 30 (b) For Bismuth 30 (c) For Uranium 30 (iv) LIST OF TABLES To Follow Page TABLE I - Lead - Fb(208) • (a) Coarse Parameter Scan 21 (b) Fine Parameter Scan 21 TABLE II - Lead - Pb(208) - Optical Model Parameter Uncertainty 21 TABLE III - Bismuth - Bi(209) (a) Coarse Parameter Scan 23 (b) Fine Parameter Scan 23 TABLE IV - Bismuth - Bi(209) - Optical Model Parameter Uncertainties . . . . . . . 2 3 TABLE V - Uranium - U(238) (a) Coarse Parameter Scan 2k (b) Fine Parameter Scan 2h TABLE VI - Uranium - U(238) - Optical Model Parameter Uncertainties 2k TABLE VII - Final Optical Model Parameters 31 (v) AC KN0WLEDGEMENT5 I would like to thank my supervisor, Dr. P.W. Martin, for his interest and encouragement in the early stages of my research. Also I would like to thank Dr. G.M. Griffiths for his advice during the later stages of this work. I would l i k e to express my appreciation to students J. Heggie and R. McFadden for their useful suggestions and continued support. I wish to thank the National Research Council for the two scholarships held during the course of my work. Also I would, like to thank my wife and my parents for their continued encouragement. 1. CHAPTER 1 INTRODUCTION Because of the intractability of dealing with the many "body problem in Physic's, i t i s general i n considering the nucleon-nucleus interaction to introduce a simplifying model which describes the interaction i n some approximate and average way. The model i s usually expressed i n terms of a minimum number of parameters required to f i t the data under consideration. Although many models have been proposed to interpret particular features of nuclear reactions, with various degrees of success, only the "Optical Model" w i l l be considered here. In this model the interaction of the nucleus with an incoming beam of particles i s represented, by an "Optical Model Potential" which assumes that the nucleus looks like a refracting and absorbing sphere to the incident beam. This model w i l l be used to interpret experimental data on the elastic scattering of k.2 MeV. neutrons from three heavy nucleij Lead, Bismuth and Uranium; and to estimate the nuclear scattering to be expected at small scattering angles. In chapter 2 the'"Optical Model" parameters are determined from a least square f i t to scattering data available in the literature, for angles larger than fifteen degrees. With these model parameters the scattering i s then calculated for smaller angles. Additional effects, not contained in the usual "Optical Model", are discussed i n chapter 3/ including Schwinger scattering, Electric Polarizability scattering, and effects due to incident neutron polarization. Using the relative cross section, normalized to unity at fifteen degrees, i t is shown that these effects alter the shape of the expected cross section at small angles. The fifteen degree point was chosen because i t i s beyond the range of the small angle scattering effects and yet at an angle where the cross section is s t i l l quite higfe^-beyond this angle the cross section f a l l s off rapidly. Being primarily concerned with the relative cross section, any effects altering only the magnitude of the differential cross section can be ignored. Thus the "Optical Model" is used to define the shape of the small angle scattering cross section, based on data beyond fifteen degrees, which would be expected in the absence of additional small angle effects. Any observed difference between the theory and experimental data taken at amall angles can then be attributed to the additional small angle effects. The History of the Optical Model In 1 9 ^ 5 , Bethe ( ^ studied the conseauences of representing the nucleus in a nucleon-nucleus interaction by a simple real potential well, having two adjustable parameters, the radius and the depth. Although the model took no account of the structure of the nucleus, i t was able to give cross sections of the right magnitude for elastic scattering. However i t also predicted broad resonances and a cross section that varied slowly with energy. In 1955 i t wag shown by (1 8) Friedman and Weisskopf that these two predictions of the model were not i n agreement with the experimental data then available. The "Compound Nucleus" model of Bohr ^  (193&) was much more successful i n dealing with the narrov, closely spaced resonances i n the cross section. According to this model the nuclear reaction takes 3. place i n two steps, f i r s t the incident particle together with the target nucleus forms the compound nucleus i n which energy is shared among a l l nucleons, then the compound nucleus decays to the f i n a l products. Because the f i r s t step implies a strong interaction of the incoming particle with the other nucleons, quite opposite to the assumption of the potential well model, the Bohr model is often referred to as the "strong-interaction model". Since the energy of the incoming particle (mostly the "binding energy) is shared, no one particle has enough energy to escape quickly. This implies a lifetime for the compound state which i s long compared to the time required for the incident particle to traverse the nucleus ( ~ 1 0 s e c ) . Further i t is assumed that the mode of decay is independent of the mode of formation, a l l "memory" of formation having been lost before decay. On the basis of the long lifetime t of the compound state and the uncertainty principle one then expects these compound states to have small energy uncertainties or widths f ~ ^7 corresponding to the observed narrow resonances. Bohr did not make a detailed model of the internal structure of these resonances. However at higher energies the "Compound Nucleus" model was shown to be inadequate by results on direct interactions, which do not proceed through an intermediate state of energy-sharing nucleons. At this point the old idea of a potential v e i l was revived when i t was found that with the addition of an imaginary component to the well, i t gave a good account of the data above 10 MeV. At the lower energies good agreement could also be attained i f the cross sections were averaged over many resonances. This idea, introduced by Ferribach, Serber, and Taylor (' ? )in 19^9* suggested that the elastic scattering of nucleons "by nuclei be compared with the scattering of a light wave by a refracting and absorbing sphere. In 1950 Pasternack and Snyder solved the Schrodinger equation for the scattering by a complex potential and LeLevier and Saxorf 1^applied the method to lower energies, showing that i t gave a good quantitative account of the main features of low energy elastic scattering. They were the f i r s t to use the term "Optical Model". When Walt and. Barshall found in 195^ that the total cross section at low energies exhibited strong and systematic broad maxima, whose position varied quite smoothly with mass number, this suggested that the scattering was essentially determined by the properties of the average nuclear matter rather than by the details of the interior structure. This new data resulted i n a revival of the potential well theory and Feshbach, Porter and ('7) Weisskopf showed that the model was able to reproduce the general over-all features for a large range of scattering data. The Optical Model describes the main variation of the t o t a l and differential cross sections with atomic weight and energy. However, even allowing the well to be complex to account for the possible formation of a compound state corresponding to absorption of the incoming particle, the Optical Model does not account for the characteristic narrow resonances associated with sharply defined energy levels of the compound nucleus since these differential cross sections depend on the details of the structure. One other feature should be mentioned, at low energies where there are only a small number of decay channels open, the compound nucleus decays by the re-emmission of a particle into the incident channel i n a significant number of cases. This i s known as Compound Elastic scattering, CE. These particles, although scattered by an entirely different process, are experimentally indistinguishable from the potential scattering, usually called Shape Elastic, SE,# Thus the experimental results inheritantly contain this compound elastic scattering whereas i t i s not included i n the Optical Model. To study this further, consider neutron scattering with X- O waves only. Then the wave function for the incident particle, when in the region beyond the nuclear f i e l d , may be written as: f(r) = Q ^'lkr _ S e!k<C) (1) L>ee equation 16*] where S is the scattering amplitude. Now S i s a function of energy, and at lower energies i t exhibits many rapid fluctuations. As S varies, so do the observable cross sections. At higher energies, these resonances crowd close together and only a smeared-out cross section i s observed. Although the individual resonances are characteristic of the particular compound nucleus involved, i f averaged out they w i l l be more characteristic of the nuclear matter i n general. Suppose we now have a nuclear potential V(r) which averages over these resonances, as in the case of the Optical Model, and which leads to a scattered wave with a scattering amplitude S . If we identify this Optical Model S with the experimental average amplitude <S> , then we have: (see Hodgson'*0) ) K- f (-- <a \(Pve = 4ke real oV <*r - j l < I - «U(S)> (3) k where dT is the measured value and c5j- is the optical model value for the total cross section. The corresponding relations for the averaged elastic and reaction cross sections may be obtained in a similar way, giving: 6. The Optical Model cross sections which do hot contain this compound elastic term are instead: Thus with the assumption that S = \S/ we have: = dF L - <cr (8) It i s usual to define the Shape Elastic cross section, d s p ' ^EL and the absorption cross section S d-^ , so that the observab quantities are then: *T ~ ^ (10) * K * ^ 6 c r (9') Since the Optical Model gives us the shape elastic and absorption cross sections ($S(_ and d A , whereas the experiments give the elastic and reaction cross sections d,ffL and 6R , there is a d i f f i c u l t y i n applying the model. However with heavier nuclei the nuclear energy levels are much closer together so that the isolated resonances do not effect the interactions too greatly. Thus, although hordering on the low energy side where the compound elastic scattering becomes more important and starts interfering with our interpretation, we w i l l make the approximation that dzc can be ignored. Previous calculations by Tripard^ 3 3^ Moore Bjorklund and Ferribach ( s ) ; Gorlov, Lebedeva, and Morosov^; Batchelor, Gilboy, and Towle^ ; and Moore and Auerbach'-3 have a l l given favourable results using this approach and thereby encourage the application of the Optical Model at this low energy. Therefore we shall use this model on the elastic differential cross sections for neutron scattering at k,2 MeV. B. Optical Model Scattering Theory Part 1 - Simple Theory In using the Optical Model for the nucleon-nucleus interaction we are representing the target nucleus by a potential well. The main contribution to the scattering comes from the real part of that well. However, as stated earlier, i n order to achieve good agreement with experiment we are forced to introduce an imaginary component. It is a f a i r approximation to consider that the imaginary part of this potential determines the absorption. But the real part also affects absorption by refracting particles towards and away from the region of greatest absorption. Thus, even though only interested i n the elastic scattering, the results depend essentially on both parts of the potential. Consequently we must work with a complex potential representing the target nucleus. I f one hopes to have any reasonable success at solving the scattering equations the f i r s t time through i t is best to make some i n i t i a l simplifying assumptions, which can be corrected for later. To begin with assume: (1) that we consider only neutron scattering, so we can neglect Coulomb interactions, (2) the target nucleus has zero spin and is spherical i n shape, (3) that we consider only l o c a l forces, and there i s no explicit momentum or isotopic spin dependence to the forces, (k) the incident neutron has no spin, (an unrealistic but not a bad assumption at low energies), (5) the incident neutron i s unpolarized, (this goes without saying so long as the forth assumption holds, however at one point we w i l l drop h and retain 5«) The task ahead is to solve the Schroedinger wave equation: Vlf(r) + 5 £ (E - V M ) ] ? r ) » 0 (11) using the Optical Model Potential: V(rV = -VRE -fV) - 'tVlM g(0 (12) [Note that the potential i s only dependent on the position <Sf the incident neutron, and not on the position of the nucleons i n the target The functions f(r) and g(r) give the radial dependence for the real and imaginary components of the well respectively. The factors VRE and. VIM give their depths. The terms E , , and ^ i n equation ( l l ) are the energy, the reduced neutron mass, and Planck's constant respectively. One can solve equation ( l l ) "by expanding the neutron wave function "fe) i n a series of Legendre Polynomials ^ : TPM - L ^ PA <««.>) (13, Substituting equation (13) into equation ( l l ) , multiplying on the l e f t by Pm/c»s te}) , and integrating over a l l angles one obtains the radial equation for angular momentum : (i>0 With the introduction of the wave number k = / 3/* and the parameter ^ = K T , we get: j ^ M * ^ - -^Mu^ - 0 to) Since the nuclear forces are short range one can assume a nuclear well cut-off ^ c , beyond which the nuclear f i e l d i s negligible and so V(r) i s set to zero i n equation (15). The remaining equation has the asymptotic form: wheref F^ (p) - ^j^if) 1 are the spherical Bessel and Neumann functions, ( ^ ( f ) = - f ^ f ) J ( s e e Schiff ( 3 o ; >, page TT) I Note that equation (l6) reduces to: which i s essentially the equation (l) quoted earlier. Also i f we put S = I this is equivalent to no interaction occurring and therefore 10, reduces to the incident wave F^(f)» Thus only the outgoing wave is modified by the interaction, The solution inside the point pc can be found by a numerical integration of (15). Applying the boundary conditions that the wave function and i t s derivative must be continuous over the boundary determines the value of . The scattering amplitude A(&) is then defined as: A(&) - a4-k 2{( +^i)(s^0 P^"*(*))}• (1T) JL=o The differential cross section i s the square of the amplitude: ^(e) = (18) and the integrated cross sections are: 4.= O (19) ^ - 5# ><<-«*^ <a>. Of course not a l l A. values contribute a significant amount and in practice we usually use a cut-off, - ^4* , beyond which ( ) - (Re(^>) is a small number 6 ~ 10 . This problem has been studied by (9) Buch, Maddison, and Hodgsen and they found empirically that krre+7l<«a. is sufficiently accurate for most purposes. ( s e e sect^o c % ) 11. Part 2 - Extended Theory Now dropping the fourth assumption, consider the neutron to have an intrinsic spin of S - Va. . This, i n effect, splits each radial equation (lJ?) into (2s+|) equations, one for each substate. Doing this w i l l allow one to calculate the polarization of the scattered particles which is another measurable quantity. Since experimental results have shown that the scattered particles are generally polarized, to account for this we must require the inclusion of a spin dependent term in our optical potential, (see Ferm*^ 5) It acts on the spin of the incident particle and therehy changes the state of i t s polarization. The simplest spin dependent potential is the real spin-orbit potential and i f we use this^equation (12) becomes: V ( 0 * - VRE - 4(r) -<-" V X M gfr) + VSR k<0 t-t ( l 2 ? ) where d is the Pauli Spin Operator (Schiff ^ °^) and h(r) is the radial dependence of the spin-orbit term, VSR is i t s depth. The total wave must now be expanded as a sum over radial, angular, and spin functions: y M where A s i s a spin function, C is a Clebsch-Gordon coefficient (Messiah^), y is a Spherical Harmonic ( S c h i f f ( 3 e > ) ) . Jjlote: the I* i s inserted to insure time reversal invariance.^j 12. With a spin particle the spin couples i n two ways to the orbita l angular momentum £ , giving two different total angular momentum values J = A + V-a. and J - A.- Va_ with -C*^ having eigen values A. and + corresponding to the two spin orientations. Substituting equation (13') into ( l l ) gives a pair of equations for each value of £ ; L E (22a) - / V S R hCp) ...{6i+i ) lu* ( i») * O ~ \ " V f ) = o ( 2 2 b ) where cVx and U^(f) are the radial wave functions for the two possible spin orientations. Using the boundary conditions,"as before, yields S^ " and. S^ and the scattering amplitudes A(6) and B(©) are calculated from: AC&) - 2fk | j ( ^ 0 h + - D +- Ptoses))} (23) B (e) = i l { ( - ) e2^ P, fe)) ? ( 2 l 0 where ' * v ^ ~ ' / ^ The differential cross section i s then given by: f^Ce) = |A(e)|z + |B(©) (25) 13. and the polarization of the scattered particles i s : P, SCtt+t where n - ' - is the perpendicular to the scattering plane, and f =s f i n a l and C = i n i t i a l . Finally, to clean up the theory a l i t t l e , the elastic scattering amplitude may he expressed i n terms of an operator where: •P0(©>* / 4 ( © ) 4 B ( e ) 6-"A (27) This operator acts on the spin wave function for the incident particle to yield the scattered wave function: (see Feshbach ^ ^ ) Then the cross section can be expressed simply as: d&' • «•«»-'• ( 2 9 ) Part 3 - Complete Theory We now dispose of the f i f t h assumption, which concerned the unpolarized state of the incident beam. From equations (27) and (29) we have, after expanding the right-hand side: % = MT +IB11 + a<Re<4'B)p..t Oo) where F0 is the incident polarization vector, (see Feshbach^^). The last term i n the above equation i s an interference term due to 11*. the i n i t i a l polarization of the incident neutrons and It vanishes i n the unpolarized case. Thus we have the complete Optical Model theory now. The numerical solution of equations ( 2 2 a ) and ( 2 2 b ) allows us to C. Form of the Optical Model Potentials So far we have not been concerned with the actual radial dependence of the Optical Model Potentials. Instead they have been l e f t simply as f ( r ) , g(r), and h(r) for the real, imaginary, and spin-orbit wells respectively. What radial forms might these functions be given? In 195^ Fermi^'3^showed that symmetry considerations require the spin-orbit forces to be zero inside the nucleus. Thus we should use a spin-orbit potential which Is appreciable only on the surface region. At high energies Fernbach, Heckrotte, and Lepore and ( 8 ) Brown have calculated the radial dependence of the spin-orbit potential by interpreting the Optical Model as a sum over nucleon-nucleon interactions. They find i t has the Thomas form: where f\ = mass of pion, and C = the speed of li g h t . If the real well radial form i s chosen as uniform throughout the interior and fa l l i n g to zero i n the surface region, then this form above w i l l give the required surface peaking. Let us use this Thomas form at the lower energy. determine the values of A(©) and B(©). These then give the differential elastic cross sections by using equation (30). (31) 1 5 . Recall that absorption is principally governed by the imaginary well and since the Pauli principle inhibits absorption inside the nucleus at low energies this suggests a surface peaking for the imaginary well too. In fact Bjorltlund and Fernbach^5^ showed that at .^1 MeV. a detailed f i t could not be obtained without a surface absorption potential as opposed to a volume absorption potential. Although a derivative type well could probably be used with equal success, we shall instead go with the Gaussian form: where Ra - r. A^3 and b> = well diffuseness. Finally, what about the real well? The only restriction imposed so far i s that i t should be uniform throughout the interior of the nucleus and that i t should f a l l to zero in the surface region. Since, as yet, there is no physical reason to prefer one form of potential over another we shall choose the Woods and Saxon^37^ model: I  -PcV) = . . . . . . ( r - R I ) (33) e_xp where R| = r ^ ^ 3 and CL - well diffuseness. Summarizing then, we have the Optical Model Potential: £•6 (34) 'rr There are seven possible parameters, the three well depths VRE, VIM, and VSR; the two widths Yr<L and ; and the two lengths a and b representing the well diffuseness. However since there exist equivalent pairs of (b, VIM) which give the same results, we shall 16. f i x the value of b at b = 1.0 fermi and l e t the parameter VIM show any variation in the imaginary well. Therefore there are actually only six adjustable parameters. D. Additional Scattering Part 1 - Schwinger Scattering In 19^ 8 Schwinger^3'^ showed that i n addition to the Coulomb and nuclear scattering there was a contribution to the scattering due to a spin-orbit interaction arising from the motion of the neutron's magnetic moment in the nuclear Coulomb f i e l d . At angles near zero degrees this normally small contribution becomes progressively more dominant. The energy of a neutron moving in an electric f i e l d E is described by: H'= />« (35) where p is the momentum of the neutron and/>Vi= I-?I3J 5" magnetons. In his 19^ 8 clculations Schwinger used a plane wave Born approximation and came up with a "Schwinger" contribution to the scattering amplitude operator. Thus the +],(©) of equation (27) becomes: ./2Ce) * +>)"*- V*—* ^ ( f ) " ^ (36) Carrying this additional term through the same calculation done i n deriving the equation (30) we have: //j |* +- IB/"2- +- A«e K • H ^ (37) IT where S ^ c ^ u * ^ c o f ( e A ) 2m c Thus, using this Born approximation method, there are two additional terms in the calculation of the cross section. In 196!l- Monahan and Elwyn^ 1^ suggested another method for calculating the "Schwinger" scattering, based on a generalization of the Born approximation. Along with the three Optical Model Potentials previously described they include a "Schwinger Potential", V , (r): v,*co* A r^C <P<'> (38) 2 m z c where - C l/ r 3 for P$ and r s i s the nuclear well ( ' / r 3 for > r s radius, \tsually taken as >s s . Thus the total potential i s : V 0 ^ = J V0p+,ca| + V S c K f o r / (39) where fc is the equivalent of the nuclear cut-off p t mentioned earlier. Since the Schwinger potential is a Coulomb potential and therefore extends beyond the nuclear cut-off radius, the Schwinger term can then be treated, as a perturbation on {v(r) for r«r c } and the fi n a l cross sections can be calculated using the program "Polecat" This program uses the phase shifts from the solution to the Schrodinger equation with [v(r) ^ * r c ] as i t s input. The procedure is nicely described by Monahan and Elwyn in their a r t i c l e . This method is more accurate than the simpler "Schwinger Approximation" 18 Part 2 - Electric Polarizability Scattering There is one more contribution to the scattering that might be considered. It too is of a Coulomb nature. Since the neutron has a charge "structure", presumably caused by a "cloud" of charged... mesons around a central core (see De Benedetti ), there can thus be an induced electric dipole moment p for the neutron caused by the electric field E of the target nucleus. If <X. is the electric polarizability of the neutron, then: f = U E This can contribute to the differential cross section, particularly at smaller angles, see Ale:ksandrov and Monahan and Elwjm> . The perturbing Kamiltonian may be taken as: (ho) where *^(r) = J l / r 1 for r £ r s \/ r y for r > rs Estimates obtained by experimental observation place the value of o< somewhere in the range of 10 to 10 cm . The hi) Monahan and Elwyn technique mentioned in the previous sectL on can be used to calculate this Electric Polarizability scattering contribution to the differential cross section. As before, the polarizability term is treated as a perturbation on where: vf.i f o r and Vpoi < r) = - <* Z 3e* f (?) . The program "Polecat" is used to do this. , 19-CHAPTER 2  THE FITTING PROCEDURE A. Introduction The Optical Model Potential as described i n equation (3b) has been used to f i t the available experimental data for the elastic scattering of neutrons. We were concerned with the scattering from Fb(208), Bi(209), and U(238) at the incident neutron energy of b.l MeV. i n the centre of mass system, since this experiment is being done here. For the f i t t i n g procedure i t w#as assumed that: (1) the polarization effect,(chapter 1, B, part 3 ) / (2) the Schwinger scattering,(chapter 1, D, part l ) , (3) the Electric Polarizability scattering,(chapter 1, D, part 2) are a l l either small effects or only significant at the small angles, © 15 degrees. Generally these assumptions are a l l true. Thus the f i t t i n g has been done with what we w i l l refer to as the "pure nuclear" Optical Model. Although the data examined was for scattering from natural samples of the three elements, rather than from the actual isotopes as was assumed, this really only poses a problem i n the case of Lead where Pb(208) is only 52$ of the Lead (35°) target composition. However, Moore has shown that the only major changes this causes are"in the size of the imaginary well depth and, to a lesser extent, the size of the spin-orbit well depth. The f i r s t of these causes primarily a magnitude sfeift in the differential cross section and the second is essentially only important at the very large backward angles. Thus we have assumed that we can overlook this question and proceed as i f the scattering is from pure isotope targets. The assumed or stated errors i n the experimental cross sections was 20. 10$ or l e a s , which means the data was quite good, providing that these estimates are c o r r e c t . In order to reduce errors caused by the energy dependence o f the O p t i c a l Model parameters, only data i n the r e l a t i v e l y narrow energy region (k.O to h.k MeV.) was used f o r the a c t u a l parameter determination. A l l c a l c u l a t i o n s f o r the f i t have been done using the program "Abacus". The X' f i t routine contained i n "Abacus" has X * d e f i n e d as: where d<4(em) and A e ^ ^ ) are the d i f f e r e n t i a l e l a s t i c cross sectfons and t h e i r errors at the centre of mass angle,.@ . M i s the number of angles at which the data i s given. The s i x parameters mentioned i n chapter 1, C were varied at f i r s t manually and then automatically t o give a minimum "X* , (the "best f i t " ) . Some of the more general features of the f i t t i n g procedure that are worth noting are: . ( l ) 4d(e) increases and the d i f f r a c t i o n pattern s h i f t s to smaller angles as VRE and ""Ve. are increased. (2) The amplitude of d<$(9) i s damped with an increase i n VIM. (3) increases and the d i f f r a c t i o n pattern s h i f t s to smaller angles as a and b are increased. (h) The spin-orb i t term decreases at large backward angles. (5) The minima and maxima are determined by the value of VRE ' • In order to show how the search progressed, the parameters used and the r e s l t i n g ^ ' s have been tabulated. In a l l cases i t i s assumed that any compound e l a s t i c e f f e c t s , as described i n chapter 1, A, could be ignored. 21. B. Lead - Pb(208) Although the Optical Model parameters depend only slowly on the energy, parameters have "been calculated for experimental data only i n the small energy region of (k,0 to b.k MeV.). In this range fas), . experimental results have been published by Walt and Beyster (1955)> Bostrom and Morgan e t . a l / 7 ^ j Okhuysen and Prud'homme^ "*'7 ;^ and Gorlov et. a l . ^ \ This data was fed into "Abacus", using a coarse parameter scan to locate the general region of minimum 'Xa",(see Table I(a)). Then a more thorough, fine scan search was done for the minimum *X*U,(see Table 1(b)). Since we wish to examine how well the Optical Model predicts the small angle scattering data, we must make sure the model's parameters are determined "by the large angle data. For this reason the data was then restricted to those points between 15 and 180 degrees. Using the "best" f i t from Table l(b) as the starting point for the large angle f i t t i n g runs, an automatic search was done with "Abacus". The results give a small improvement in the % *~ , going from the i n i t i a l value of 10.1--8 to the best value of 10.J+3 when VRE a 1+6.29 MeV. and a =. 0.68 fermi, with a l l other parameters remaining Unchanged. This then gives the "best" parameters from the large angle data and we w i l l use these to predict the small angle scattering. From the values i n Table l(b) we can get an idea of the accuracy to which these "best" parameters are specified. A 2.2$ change in the value of VRE near the minimum "X1 position results i n a 3^$ change i n the value of 'Xx . We shall draw the limit at a Vfo change i n "X*", saying that with anything less i t is entirely unreasonable TABLE I (a) LEAD - Fb(208) - COARSE P A R A M B m j C A H j , *NOTE* r r e = 1.2k fm. - 1.21+ fm. a = O.67 fm. b = 1.0 fm. VRE(MeV) VIM(MeV) VSR(MeV) hk.O 3.0 6.0 20.83 8.0 21+.03 10.0 28.36 5.0 6.0 M+.61 8.0 1+6.22 10.0 1+7.52 T.o 6.0 61+.9S 8.0 65.28 10.0 61+. 88 k5.1 3.0 6.0 16.11 8.0 15.71 ' 10.0 18.30 5.0 6.0 32.11+ 8.0 33.32 10.0 35.00 7.0 6.0 5L-+5 8.0 52.35 10.0 52.98 kj.0 3.0 6.0 15.U8 8.0 12.30 10.0 13.26 5.0 6.0 21+.99 8.0 25.97 10.0 28.01 Minimum Region: 1+7.0 3.0 8.0 12.30 TABLE I (*>) ; LEAD -Pb(208) ~ FINE PARAMETER SCAN *N0TE* b = 1.0 fm. VRE(MeV) VBl(MeV) VSR(MeV) r„(fm) r^Cfm) a(fm) 1*8.0 2.0 8.0 1.23 1.23 .67 ll * . 6 8 1*7.0 2.0 1.2k 1.2k 12.97 3.0 12.30 1.26 13.58 1.22 11.51 ^6.5 2.9 8.5 1.21* i . 2 l * .65 15.65 .67 13.^5 .69 12.27 1.19 .65 11.1*7 .67 10.1*6 .69 10.70 1*5.5 2.86 8.5- 1.21* 1.19 .67 ll * . 5 0 1*6.5 10.1*6 1*7.5 13.29 1*7.0 2.85 8.5 1.21* 1.19 .67 10.81 1.21 10.88 1*6.5 2.86 8.5 1.21* 1.19 .65 11.1*7 .67 10.1*6 , .69 10.70 1+6.5 2.80 6.8 1.21* 1.19 .67 IO.65 2.80 8.5 10.3I* 2.92 10.61 2.86 10.1*6 2.80 10.31* 2.71* 10.23 2.55 10.00 2.1*1* 9.98 2.35 10.00 Minimum: 1*6.1*6' 2.1*1* 8.5 1.21* 1.19 .67 9.98 TABLE II LEAD -Pb(208) - OPTICAL MODEL PARAMETER UNCERTAINTY PARAMETER $ VARIATION IN PARAMETER VALUE AND $ ERROR IN PARAMETER PARAMETER ERROR oYe) VRE 2.2$ 3>*$ 1*6.29 i .03 MeV. (.07$) .85$ .17$ VTM 1*.1$ .2$ 2.14-1+ t .1+8 MeV. (20.$) .75$ .50$ VSR 20$ 3$ 8.5 * .6 MeV. (7$) 1.15$ .50$ a 3$ 6.5$ 0.68 t .003 fm. (.1*6$) .07$ .8$ 1.7$ 1.2k t .006 fm. (.1*7$) 1*.0$ 1.0$ r , 1.5$ ' .7$ 1.19 * .02 fm. (2$) .60$ .19$ Total Error l n Cross Sections: t 7.7$ i 2.1*$ 22. to c a l l one Kx better than another. Even at 1$ this i s d i f f i c u l t to judge. However, allowing this 1$ definition on "X7" means that TOE is specified t o ( a ' % * / ) ^ = ' o 7 % * Thus we have: TOE = 1+6.29 ± .03 MeV. (.07$) Similarly the errors on the other parameters have been determined and are shown i n "Table II. Looking at the Optical Model prediction at two degrees and also at fifteen degrees and allowing this .07$ variation i n TOE ve have a ± . 8 5 $ uncertainty i n ^/da. but only a ±.17$ uncertainty i n the relative cross section ffftysi)r±[. ' This is due to the fact that the parameter variation causes essentially onlj r a magnitude shift i n the differential cross section which does not show iip i n the relative cross sections. This type of calculation was done for each parameter, as show i n Table II. It i s assumed here that each parameter acts independently and thu3 a two parameter simultaneous variation would cause an error equal to the sum of the errors caused hy the two variations done separately. Although this is not generally correct, the result w i l l be a reasonable estimate of the actual uncertainty involved. In thi3 way we have placed an estimate on the error involved in the Optical Model prediction i t s e l f . This estimate, is ewcirely dependent on the IC" calculation and is associated only with the value of X 1 and the least squares f i t t i n g procedure. In the case of Lead, the Optical Model prediction for the relative differential cross section is determined to within an uncertainty of only 2.1+$. r ., The "best f i t " prediction is shown, along with the experimental data, i n Figures l(a), .anfi; l ( b ) . Although the predicted cross sections FIGURE IL (a) •H *H -P C CD 0^ o M E--: U PM co CO CO o K o o H E-' CO H R 10.0 _ 9.0 8.0 7.0 6.0 DIFFERENTIAL ELASTIC CROSS SECTIONS FOR LEAD -Pb(208) 5.0 k.O 3.0 2.0 1.0 ° OPTICAL MODEL PREDICTION * WALT & BEYSTER (1955) a G0RL0V ET. AL. (1967) A B0STR0M, MORGAN, ET. AL. (1959) 7 0KHUYSEN, & PRUD'HOMME (1959) 0.0 ± k0 60 80 -100 120 IkO l6d SCATTERING ANGLE Q (CM.) (degrees) 180 i i 1 FIGURE 1 (b) 10.-0T ' i r — 1 - ! i [ ~ ~ T "1 ! j DIFFERENTIAL ELASTIC CROSS SECTIONS 1.0- V OKHUYSEN & FRUD'HOMME (1959) o.ol 1- • I , 1 , 1 , I , i . i . ! ' 0 5 io 15 20 25 30 35; ! SCATTERING ANGLE 9 ( c.M.)(degrees) \ i n Figure l(b) look quite high compared to the experimental data, the discrepancy is of a magnitude nature only. Thus the 7.7$ estimated uncertainty i n the Optical Model cross section at small angles easily absorbs this apparent error. And since the data below 15 degrees, where most of this discrepancy occurs, i s a l l from the one reference; Gorlov et. a l . ( which gives very l i t t l e information concerning the techniques of their measurements)^ we are s t i l l very confident about the vali d i t y of the theoretical prediction. In fact, one can see that the shape suggested by the experimental data nicely follows that predicted by the Optical Model curve, certainly to within the estimated 2.k$> uncertainty in the relative cross section, except perhaps below five degrees. However at this point we can no longer ignore the small angle corrections and one would, expect a departure from the theoretical "pure nuclear" curve. A plot of the relative cross section i s shown in Figure k» . Bismuth - Bi(209) Using the data of Walt and Beyster^ ; Gorlov e t . a l . ; Thomson^31 ^  ; and Ueddell^ 3 6^ exactly the same procedure was followed for Bismuth as was used for Lead. The coarse scan search is shown in Table III(a) and the fine scan search i n Table I l l ( b ) . Again using the minimum found i n Table Ill( b ) as a starting point, the data was restricted to the 15 to 180 degree region and an automatic search was done on the parameters. The K value changed frem 12.31 to the f i n a l value of 10.56. This last value was obtained with TABLE III (a) BISMUTH - Bl(209) - COARSE PARAMETER SCAN *NOTE* = ro a = O.65 fm. b = 1.0 fm. VRE(MeV) VIM(MeV) VSR(MeV). r 6 (fm.) 1+8.0 3.0 5.0 1.27 58.82 8.0 IA.03 11.0 23.07 5.0 5.0 29.67 8.0 2U.35 11.0 21.97 7.0 5.0 31.60 8.0 31.28 11.0 33.19 50.0 3.0 5.0 1.25 59.02 8.0 5^.25 11.0 5.0 5.0 39.7-+ 8.0 35.81 11.0 36.91 7.0 5.0 1+6. kg 8.0 U6.97 11.0 50.U7 52.0 3.0 5.0 1.23 7^ .80 8.0 60.87 11.0 55.13 5.0 5.0 60.77 8.0 57.M 11.0 60.73Minimum Region: 1+8.0 5.0 11.0 1.27 21.97 TABLE III (b) BISMUTH - Bl(209) - FINE PARAMETER SCAN *NOTE* b» 1.0 fm. VRE(MeV) VIM (MeV) VSR(MeV) r0(fm.) a(fm.) 1+8.0 5.0 11.0 1.27 .65 21.97 1*5.0 10.0 1.29 17.08 "+3.0 11.9k 1+5.0 6.0 18.85 1+7.0 k.5 1.28 18.99 12.0 16.65 kl. 0 k.5 10.0 1.295 .65 28.67 1.28 18.99 1.265 1U.03 1.26 13.30 U6.5 •4.3 10.0 1.26 .65 12.I+7 .63 12.92 .67 12.50 5^-5 k.3 10.0 1.26 .65 12.76 U6.5 12.1+7 Vf.5 13.9»+ 1+6.5 k.2 10.0 1.26 .65 12.21+ l+.i 12.01+ 3.T 12.0 11.60 3.63 11.59 3.5 11.62 U6.5 k.2 ll+.O 1.26 .65 12.77 12.0 12.07 10.0 12.2k 8.0 15.87 Minimum: 46.5 3.63 12.0 1.26 .65 11.59 TABLE IV -BISMUTH - Bj ( g Q 9 ) - OPTICAL MODEL PARAMETER UNCERTAINTIES PARAMETER $ VARIATION IN PARAMETER VALUE AND $ ERROR IN PARAMETER 'X2 PARAMETER ERROR efCB) VRE 2.2$ 7$ 1*5.61* t .86 MeV. (.63$) .12$ .01*$ VIM 2.8$ .17$ 3.63 ± 1.2 MeV. (33$) .28$ .05$ VSR 20$ 15.5$ 12.0 ± .31 MeV. (2.6$) .1*2$ .11*$ a 3$ 2$ 0.70 ± .02 fm. (3$) .84$ .52$ r Q .8$ 1.7$ 1.26 * .01 fm. (.<*$) .1*3$ .20$ Total Error in Cross Sections: 2.09$ ±.95$ FIGURE :g (a) 10.0 DIFFERENTIAL ELASTIC CROSS SECTIONS 9.0 FOR BISMUTH - Bl(209) 8.0 __ 7.0 6.0 5.0 h.o 3.0 2.0 1.0 0.0 o OPTICAL MODEL PREDICTION * WALT & BEYSTER (1955) ES GORLOV ET. AL. (1967) A BOSTROM, MORGAN, ET. AL. (1959) • THOMSON (1963) B WEDDELL (1956) 1*0 60 80 100 120 1X0 SCATTERING ANGLE e (CM.) (degrees) FIGURE Vg. (b) 10.0 c • H 4) •P «3 C SH ti , 0 3 O M E-< a CO CO CO o O CO M E-i fe M P SCATTERING ANGLE 8 (C.M.)(degrees) VRE = " +5MeV. and a = 0.70 fermi. The other parameters remained the same. These new parameters are then the "best" parameters as determined by the large angle data. Again we can get an idea of the accuracy of the parameters by examining the resulting change in "X*8" for prescribed changes in each parameter. Since only about one half as much data was available for Bismuth as compared to that for Lead, the value of TC^can not be defined as well. Hence we a r b i t r a r i l y draw the limit on " X * to 2$, Saying one set of parameters is better than another i f there is more than a 2?3 difference i n their respective X * . These calculations and the resulting cross section uncertainties are tabulated i n Table IV. Adding up the errors as before we estimate that the relative cross section i s determined to within an uncertainty of only 1$. The "best f i t " prediction, along with the experimental data, i s shown i n the Figures 2(a) and 2(b). Again the apparent discrepancy is a magnitude shift only and the data below 15 degrees is a l l from the same reference. The shape is well predicted as i n the case of Lead and Figure h shows the predicted relative cross section at small angles. D. Uranium - U(238) For Uranium an Optical Model f i t was done to the data of Gorlov e t . a l . ; Batchelor, Gilboy, and Towle^ j and Walt and Beyster^3S"^. The coarse and fine scan searches are shown i n Tables V(a) and V(b) respectively. Using the minimum "XXset of parameters TABLE V (a) URANIUM - U(238) - COARSE PARAMETER SCAN *N0TE* rVe - r lm ~ r o a = O.65 fm. b = 1.0 fm. VRE(MeV) VBl(MeV) VSR(MeV) r 0(fm.) 50.0 b.o 6.0 1.25 11+5.62 6.0 111.1+2 8.0 1+8.0 U.o 68.37 6.0 1+8.37 8.0 1+3*00 1+6.0 1+.0 35.85 6.0 16.50 8.0 13.66 1+6.0 1+.0 6.0 1.27 U7.86 6.0 28.1+1+ 8.0 23.69 1+.0 1.29 72.00 6.0 1+8.09 8.0 39.36 1+6.0 6.0 8.0 1.25 1U.11+ 1.27 25.87 8.0 1.25 12.78 1.27 22.11+ 1+6.0 It-.O 10.0. , 1.25 2I+.85 6.0 11.87 8.0 12.06 Minimum Region: 1+6.0 6.0 10.0 1.25 II.87 TABLE Y M URANIUM - U(238) - FINE PARAMETER SCAN *NOTE* r r a - r C m * r 6 b = 1.0 fm. VRE(MeV) VIM(MeV) VSR(MeV) r<> (fm. ) a (fm.) X1 •+3.5 6.5 12.0 1.21+ .65 16.51 Mu5 11.45 W+.5 6.5 10.0 1.25 .65 9.81 12.0 8.87 10.0 1.21+ 11.1+5 U5.22 6.65 10.0 1.25 .65 8.62 .63 9.12 .67 8.26 .68 8.16 .69 8.11 .70 8.13 1+1+. 22 6.65 10.0 1.25 .69 9.11 1+6.22 13.-+2 1+5.22 6.80 8.0 1.25 .69 9.60 10.0 7.89 12.0 7.68 7.20 12.0 7.55 7.39 7.52 7.60 7.56" Minimum: 1+5.22 7.39 12.0 1.25 .69 7.52 TABLE VI URANIUM - U(236) - OPTICAL MODEL PARAMETER UNCERTAINTIES PARAMETER $ VARIATION IN PARAMETER VALUE AND $ ERROR IN PARAMETER X1 PARAMETER ERROR 6(e) «•«/. VRE 2.2$ 39$ 1*1*.80 * .13 MeV. (.28$) .38$ .19$ VIM 2.7$ .1*9$ 7.39 ± 2.0 MeV. (27$) 5.1+5$ .10$ VSR 20$ 12.0 ± 2.2 MeV. (19$) .98$ .17$ a 1.5$ .1*3$ O.69 * .11 fm. (17$) 1.0$ .50$ 0.8$ 13.8$ 1.25 t .OOU fm. (.29$) .^ 5$ .20$ Total Error i n Cross Sections: ±8.26$ ±1.16$ 25-from Table v(b), an automatic parameter search was done on the data i n the range 15 to 180 degrees only. The K^vent from a value of 7.29 down to 6.76 which occurred at VRE = 1}.1*#80 MeV. with a l l the other parameters unchanged. This then is the "best" set of parameters as" determined from the large angle data. Table VI gives the error estimates on the parameters, assuming an arbitrary 5$ limit on the value of 'X**. This higher figure was chosen for "X1 because there i s even less data available for Uranium at this energy and that which i s available clearly has wide deviations i n i t . The relative cross section uncertainty i s also given i n Table VI with a f i n a l value of 1.16$ being put on i t . The graphs for the Optical Model predictions and the experimental data points are shown in Figures 3(a) and 30b). The relative cross section at small angles i s shown plotted i n Figure 1*. FIGURE 31(a) ' 10.0 9.0 L i -8.0 7.0 T T THE DIFFERENTIAL ELASTIC CROSS SECTIONS  FOR URANIUM -U(238) O OPTICAL MODEL PREDICTION * WALT & BEYSTER (1955) V BATCHELOR, GILBOY, & TOWLE (1965) 6.0 5.0 l+.O-3.0-FIGURE" o.o ., I — i -I 1 " - i i i • i • l _ t I 0 5 10 15 20 25 30 35 SCATTERING ANGLE Q (CM.) (degrees) RELATIVE CROSS SECTION 4<a)rmL (normalized to unity at 15°) 26. CHAPTER 3 CORRECTIONS TO THE OPTICAL MODEL As mentioned at the beginning of chapter 2, the f i t t i n g procedure was carried out ignoring effects due to Incident Polarization, Schwinger scattering, and Polarizability scattering. The Optical Model predictions obtained so far are referred to as the "pure nuclear" cross sections. Thus Figures 1 through h show the "pure nuclear" curves. Since the three effects l i s t e d above are small and primarily at small angles only, and since the pure nuclear curves were obtained, by using the large angle experimental data only, then these curves are indeed a valid prediction for this "pure nuclear" part of the scattering at small angles. Thus a l l that remains i s to include the above three effects along with the "pure nuclear" part. This i s what has been done i n this chapter. A. Incident Polarization Effects The experiment being done here uses incident neutrons that are obtained from the reaction X*(d,n)He3 . This reaction produces a par t i a l l y polarized neutron beam and this incident polarization must be talcen into account when annalysing the experimental data. Dubbeldam and Walter ^  have measured the polarization of neutrons being produced at k$ degrees i n the laboratory frame of reference for a deuteron energy of 1.9 MeV. They obtained a value of 15.4 $ (± 1.7$). Since the experiment being done here has a similar physical set up, we shall use this value of 15.4$ i n the calculations of o cfl •H •e cd u <u -p m C 03 ^3 at 111 O H EH O pa co co CO o PC o E - i CO H o 5.5 5.0 h.5 U.o 3.5 3.0 2.5 2.0 1.5 1.0 0.5 -II. 0.0 — i 1 r - — i 1 r FIGURE 5:: (a) 1  INCIDENT POLARIZATION EFFECTS FOR SCATTERING FROM LEAD Curve Incident Polarization 1 , 1 , 1 1*0 60 80 100 120 ll+0 160 180 SCATTERING ANGLE e (c.M.)(degrees) 'SCATTERING ANGLE 6 (CM.)(degrees) 27. the Incident Polarization effects. Although "Abacus" normally calculates the di f f e r e n t i a l cross section using equation (25)., we have modified the program so that i t w i l l use equation (30) instead. "Abacus" was then run using the "best f i t " parameters determined i n chapter 2. The results of these calculations for Lead are shown in Figures 5(a) and 50b). The effect i s indeed quite small, as expected, except i n the arelbeyond 155 degrees. However, since there was very l i t t l e data i n this region for the purpose of f i t t i n g , the "pure nuclear" prediction i s s t i l l valid. At the small angles the effect gives about a 1.5$ change in the cross section. However this i s essentially a magnitude change and the relative cross section curves for the two possible polarization directions are indistinguishable at small angles from the "pure nuclear" curves i n Figure h» Thus i t might appear that we can ignore this effect altogether, but i t w i l l prove more important later, after we discuss the Schwinger scattering. B. Schwinger Scattering At very small angles, Schwinger scattering makes a significant contribution to the cross section. In chapter 1,B two different methods were discussed for the inclusion of this scattering. We have modified "Abacus" so as to calculate the differential cross section using the "Schwinger approximation" method,., equation (37)» In addition the Monahan and Elwyn method was also used. In order to do this "Abacus" was again modified so that i t would include the potential of equation (38) along with the real, imaginary, and spin-orbit 10.0 c •rl a u 0) •p to C o H CO CO CO o PC o f-1 w M p 10 15 20 25 SCATTERING ANGLE Q (CM.)(degrees) potentials. The output phase shifts from "Abacus" were then fed into the program "Polecat" which i n turn calculated the d i f f e r e n t i a l cross sections. The results of the calculations using these two methods are shown, for Lead, in Figure 6. The other nuclei give similar results. Although an incident polarization of zero has been assumed in order to do these calculations, this w i l l be r e c t i f i e d later i n part D. From the graph i t can be seen that this effect is very small at anything larger than 15 degreest The two methods are nearly equivalent, especially as far as the relative cross section i s concerned. The Monahan and Elwyn approach, however, does include a 3mall amount of additional scattering that the "Schwinger approximation" leaves out. It is also clear from the graph that this Schwinger scattering effect cannot he ignored when the cross section is desired for angles less than 10 degrees. . Electric Polarizability Scattering In order to calculate this contribution to the scattering, "Abacus" was modified so that i t would include the potential of equation (hi). Then, using the "best f i t " parameters for each nucleus, programs "Abacus" and "Polecat" were used, in turn to generate the "pure nuclear plus polarizability" cross sections. The results for a l l these nuclei are roughly the same and so only the Lead plots are shown. The calculations were done assuming three different values for the polarizability. Values of ol - 10 cm. -n 3 and oL — 10 cm. give results which are indistinguishable and are 'i° -a shown as the one curve B i n Figure 7. The case where * - 10 cm.9 29. which i s considerably larger than the theoretical estimates of -a. x 10 cm., shows a slightly larger effect, curve C. However, even so, the result i s primarily a magnitude change which even at the small angles i s less than a ltfo contribution. Thus for the determination of the shape of the cross section at small angles, the "polarlzabllity" term could be ignored, however for the sake of completeness, and since i t i s not really any more work to do so, we shall include this effect when calculating the f i n a l "corrected" Optical Model prediction. We -HI A shall use a value of «* ~ 10 cm. for that calculation since i t i s a more reasonable value in terms of the theoretical estimates. D. Final Corrected. Optical Model Prediction We shall now combine a l l of the above terms and calculate the f i n a l Optical Model prediction of the cross section. Including the Polarization effects with the Schwinger scattering w i l l cause us to pick up the additional Schwinger term i n equation (37) that depends on the i n i t i a l polarization of the incoming neutrons. This can enhance the Schwinger scattering at small angles and makes i t even more imperative that these terms be included.. Using a polarlzabllity of °* 10 cm. and the incident polarization figure of 1 5 . , the f i n a l results for the three nuclei are shown in Figures 8(a), 8(b), and 8(c). Notice that for the negative incident polarization case, (the direction convention was given in equation (26)), the two Schwinger terms of equation (37) almost cancel out except for below three degrees. The positive 15»k$> incident polarization case has a much stronger Schwinger contribution and i s clearly visable at 15 degrees where i t FIGURE 8 (a) SCATTERING ANGLE €» (CM.) (degrees) FIGURE :8 ^ 3 0 5 10 15 20 25 30 35 SCATTERING ANGLE & (CM. ) (degrees) FIGURE .fc.f<}) SCATTERING ANGLE e (c.M.)(degrees) 30. i s s t i l l a .3*5$ correction. These effects significantly change the shape of the small angle cross section a 3 shown i n Figures 9(a), 9(b), and 9(c). The relative cross section i s altered "by approximately 3$ at five degrees and 6$ at three degrees for the P0 = +15.V$ case, and thus should he detectable by the experiment being done here, which uses the positive polarization geometry. 2.0 FIGURE 9 (a) T 1 —1—' 1.8 FINAL OPTICAL MODEL PREDICTION FOR THE RELATIVE DIFFERENTIAL CROSS SECTIONS FOR LEAD H -p GJ >= -P • H a o -p o x> o H EH O W CO to CO o O EH < o.i* 0.2 Curve A - Pure Nuclear Prediction B - Corrected Prediction, PD= -15.1*$ C - Corrected Prediction, Pc= +15.1*$ 0.0 i 8 12 16 20 SCATTERING ANGLE e (CM.)(degrees) 21* 28 2.0 FIGURE (h) 1.8 o LP H -P 1.6 • n C o -p OJ • H H o o H L"H o c-: to CO CO o o EH 1.4 — 1.2 1.0 0.8 0.6 FINAL OPTICAL MODEL PREDICTION FOR THE RELATIVE DIFFERENTIAL CROSS SECTION FOR BISMUTH Curve 0.4 _ A - Pure Nuclear Prediction B - Corrected Prediction, P0= -15.4$ C - Corrected Prediction, P 0 a +15.4$ 0.2 0.0 1 8 12 16 20 24 SCATTERING ANGLE 6 (C.M.)(degrees) 28 FIGURE :9T (c) 2.2 1 i 1 1 i | i | . | i | 1 - \ FINAL OPTICAL MODEL PREDICTION -2.0, -— 1 FOR THE RELATIVE DIFFERENTIAL CROSS SECTIONS — FOR URANIUM -1.8 — 1 — 1.6 - y — l.l* • — 1.2 — 1.0 — — 0.8 — 0.6 Curve A - Pure Nuclear Prediction AfcB \> — o.i* B -C -Corrected Prediction, PQ = -15.1*$ Corrected Prediction, Pe = +15.1*$ 0.2 I , 1 1 . 1 . 1 1 1 , i 1 0 U . 8 12 16 20 2h _ 28 SCATTERING ANGLE 9 (CM.)(degrees) 31. CHAPTER k  DISCUSSION AND CONCLUSIONS For the purposes of easy reference, the "best f i t " parameters for a l l three nuclei have been l i s t e d again in Table VII. A look at these values w i l l show that the real well depth and radius are quite accurately determined for Lead and Uranium and f a i r l y well known for Bismuth. This accuracy is due to the fact that the real well gives the main contribution to the elastic scattering. The imaginary well, being responsible for absorption, is of less importance and correspondingly is not as well specified. The errors are considerably larger for the imaginary well depth. The spin-orbit well was necessary but of less importance than the real well and so i t too i s not as accurately known. It is specified most poorly for Uranium, primarily because there was not sufficient data available at the large backward angles. The real well diffuseness is known to a good accuracy for Lead, and to a lesser degree for Bismuth and Uranium, again probably due to the poorer amount of data available. With the errors in the parameter specification we must accept an error i n the predicted values of the cross section. In magnitude of the cross section this error could be about 7$ in the case of Lead, 8$ for Uranium, and possibly as low as 2$ for Bismuth. Since a l l of the parameters of Table VII were determined from the large angle data, i t is of interest to examine how well the predictions f i t at the smaller angles. As Figures 1, 2, and 3 show, the f i t is f a i r l y TABLE VII  FINAL OPTICAL MODEL PARAMETERS *Note* b = 1.0 fm. LEAD - Pb(208) VRE = 46.29 t .03 MeV. (.07$) VIM = 2.44 * .48 MeV. (20$) VSR = 8.5 ± .6 MeV. (7$) a = 0.68 i .003 fm. (.46$) * V = 1.24 i .006 fm. .(.47$) = 1.19 t .02 fm. (2$) BISMUTH -Bl(209) VRE = 45.64 ± .86 MeV. (.63$) VIM " 3.63 t 1.2 MeV. (33$) VSR = 12.0 t.31 MeV. (2.6$) a = O.TO t .02 fm. (3$) To = 1.26 * .01 fm. (.94$) URANIUM -U(238) VRE = 44.80 i .13 MeV. (.28$) vm = T.39 i 2.0 MeV. (27$) VSR = 12.0 t 2.2 MeV. (19$) a = 0.69 i .11 fm. (17$) = 1.25 * .004 fm. (.29$) 32. good at the larger angles. In the small angle region, the predictions 3'e.em to be out i n magnitude. However If one considers the 2 to 8 $ error mentioned above, and. the error bars associated with the experimental values, the discrepancies are accounted for. The predicted curve follows the shape of the experimental points very well down to about five degrees, certainly to within the 1 to 2 $ estimates on the uncertainty of the relative cross section prediction. The data for a l l three nuclei seems to show a possible divergence from the predicted shape below five degrees. This could be the Schwinger scattering. However there is Insufficient, data here to draw any conclusions. Figures 9(a), 9(t>)> and. 9(c) show clearly that Schwinger scattering should be observed at the small angles. Careful study at five degrees or less should reveal the Schwinger scattering for the case of the positively polarized incident beam of neutrons. I f the negatively polarized beam is used, the measurements would have to be at less than two degrees for any reasonable hope in showing the Sbhwinger contribution. Figure 7 shows that i t is f u t i l e to attempt a determination of the polarlzabllity °( by examination:of differential cross section data at this energy. The polarlzabllity scattering effect i s just too small to be seen. 33. CHAPTER 5 REFERENCES 1. Abacus - Auerbach, E.H., Brookhaven National Laboratory Report BNL 6562, (unpublished I962) 2. Aleksandrov, Y.A. and Bondarenho, I.I. - Zh. Eksperim. i Teor. Fiz, 31, 726, (1956), English translation, Soviet Phys. JETP, 17, 89, (1963) 3. Batchelor, R., Gilboy, W.B., and Towle, J.H. - Nuclear Physics, 65, 236, (1965) I*. 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