T H E U N I V E R S I T Y OF M I C H I G A N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Physics Technical Report No. 25 PROTON-DEUTERON ELASTIC SCATTERING AT HIGH MOMENTUM TRANSFERS Ernest Coleman ORA Project 03106 under contract with: DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH WASHINGTON, D.C. CONTRACT NO. Nonr-1224(23) NR-022-274 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1966

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1966.

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii ABSTRACT ix I. INTRODUCTION 1 II. ONE-NUCLEON EXCHANGE MODEL 3 A. Proton-Neutron-Deuteron Vertex Function 5 B. Vertex Form Factors 7 Co Differential Cross Sections 10 III. THE THREE-NUCLEON SYSTEM 21 IVo OFF THE FORWARD DIFFRACTION PEAK 32 A. Impulse Approximation 32 B. Multiple Scattering Processes 37 V. EXPERIMENTAL PROCEDURES 45 A. Beam 45 B. Beam Optics 46 CO Liquid Deuterium Target 46 Do Determination of Scattering Angles 50 E. Detection of Events 52 VI. NORMALIZATION OF CROSS SECTIONS 56 VII. DATA CORRECTIONS 59 A. Nuclear Absorption 59 B. Counter Efficiency and Dead Time 60 C. Beam Attenuation 61 Do Angular Uncertainty 61 E. Multiple Coulomb Scattering 62 F. Accidental and Background Coincidences 62 VIII. RESULTS OF EXPERIMENT 66 IXo CONCLUSIONS 72 iii

TABLE OF CONTENTS (Concluded) Page APPENDIX Ao EVALUATION OF MATRIX ELEMENT 75 APPENDIX B. DEUTERON WAVE FUNCTIONS 77 BIBLIOGRAPHY 81 ACKNOWLEDGMENTS 86 iv

LIST OF TABLES Table Page I. Magnet Currents 48 II. Magnet Characteristics 48 IIIo Counter Dimensions 53 IV. Cross Section for C12(p,pn)C11 58 V. Uncertainty in Scattering Angles 63 VI. Backward Proton-Deuteron Elastic Scattering 67 VII. Forward Proton-Deuteron Elastic Scattering 67 VIII. Coefficients in Exponential Fit 70 70

LIST OF FIGURES Figure Page 1. Notation for center-of-mass system and one-nucleon exchange 4 Feynman diagram. 2. Form factor from Hulthen wave function with core of various radii. 9 3. Form factors for employing various wave functions. 11 4. Unnormalized theoretical differential cross sections at 1.0 GeV. 13 5. Theoretical cross sections at 1.0 GeV. 14 6. Theoretical cross sections at 1.3 GeV. 15 7.- Theoretical cross sections at 1.5 GeV. 16 8. Theoretical cross sections at 0.660 GeV. 17 9. Theoretical cross sections at 0.715 GeV. 18 10. Theoretical cross section at 3.66 GeV. 19 11. Diagrams for proton-deuteron scattering. 24 12. Unnormalized Born terms at 1.0 GeV. 26 13. Contribution of Born terms at 1.0 GeV. 27 14. Contribution of Born terms at 1.3 GeV. 28 15. Contribution of Born terms at 1.5 GeV. 29 16. Contribution of Born terms at 0.660 GeV. 30 17. Prediction of impulse approximation at 2.0 GeV. 36 18. Diagrams for single and double scattering. 39 19. Effect of the real parts of the nucleon-nucleon scattering amplitudes on the proton-deuteron differential cross section 43 vii

LIST OF FIGURES (Concluded) Figure Page 20. Theoretical differential cross section in the laboratory system when single- and double-scattering interaction are included. 44 21. The beam layout. 47 22. Plan view of experiment. 49 230 Current in bending magnet H206 vso cos 6* at various energieso 51 24, Electronic logic circuitry. 54 25. Experimental differential cross sections at 1.0, 1.3), and 1.5 GeV, 68 26, Experimental differential cross sections at 2.0 GeV. 69 27. Exponential fit to cross sections at 1,0, 1.3, and 1.5 GeV. 71-7. 28. The effects of relativistic and nonrelativistic propagators at 1.0 GeV. 74 *i *. 1

ABSTRACT An experimental and theoretical investigation of proton-deuteron elastic scattering at high momentum transfers is presented. The differential cross sections for backward elastic scattering at incident proton kinetic energies of 1.0, 1.3, and 1.5 GeV have been measured for four-momentum transfer squared (-t) from 2.6 to 5.0 (GeV/c)2, which corresponds to cosine of centerof-mass proton scattering angles (cos G*) from -0.5 to -0.9. A backward peak is observed, and the slope and magnitude of the peak have been determined. At 2.0 GeV for forward elastic differential cross section has been measured for -t from 0.44 to 1.54 (GeV/c)2 qf cos G* frgm 0.875 to 0.565. A shoulderlike departure from the forward diffraction peak was observed. The one-nucleon exchange peripheral model has been successful in interpreting the backward peak. Calculations based on modern three-body quantum mechanical formalisms for the three-nucleon system supporting a two-body bound state also suggest the one-nucleon exchange process as the dominant mechanism. The measured forward differential cross section has been explained by the importance of double-scattering of the incident proton at higher momentum transfers.'A negative value for the ratio of the real part to the imaginary part of the neutron-proton elastic scattering amplitude at 2.0 GeV is shown to yield maximum agreement with the experimental data. ix

I. INTRODUCTION The interaction between two nucleons has long been the fundamental problem in particle physics. The deuteron is the only bound state of the twonucleon system, and the elastic scattering of protons by deuterons provides an opportunity to examine any theory which proposes to describe the static and dynamical properties of the two-nucleon system. The static properties of the deuteron have been known for many years. The spin of the deuteron was determined by Murphy and Johnstonl to equal one 1i unit. The magnetic moment of the deuteron was first measured by Stern and Estermann,2 who deflected deuterium molecules in an inhomogeneous magnetic field. A more accurate measurement using the nuclear magnetic resonance absorption method was made by Widmett,3 who reported the ratio of the deuteron magnetic moment to the proton magnetic moment thus yielding I'd = 0.857411 + 0.000019 nuclear magnetons (nuclear magneton = ef/2mc where m is the proton mass and e is the proton charge). The deuteron electric quadrupole moment was postulated and measured by Kellogg, Rabi, and Ramsey who noticed a discrepancy between their measured deuterium radiofrequency magnetic resonance spectrum and theory. The experimentally measured fine structure was much larger than the magnitude predicted by thirdrorder perturbation calculations. Using an improved design of the molecular beam apparatus, Kolsky et al.5 found the quadrupole moment, Q = 0.2738 f2 (f = 10-13 cm). In order to explain the presence of the electric quadrupole moment Schwinger6 proposed a tensor contribution to the nuclear force, which gave the deuteron a 3S1 + 3D ground state. In the low energy scattering problem extensive use is made of the effective range theory. This method assumed that the scattering cross section could be expressed in terms of only the S-wave phase shift, and such quantities as the triplet scattering length at = 5.41 f, the singlet scattering length as = -23578 f, and the effective range re = 1.69 f were determined. However in high energy scattering, the assumptions of the effective range theory are no longer valid. Theories must then be tested by computing the differential cross sections and comparing them with the experimental data. In the 100 MeV to 700 MeV incident proton kinetic energy range, several determinations of the differential cross sections have been made,7-13 The common characteristic of these measurements was a strong forward peak and a smaller backward peak. No other structure was seen at these energies. Above 700 MeV, protondeuteron elastic scattering data are very scarce. Bayukov et al}4 have measured a single point of the differential cross section near 1800 at three energies) 0.715 GeV, 1.00 GeV, and 3.66 GeV. Kirillova et alo15 have determined the differential cross section for angles less than 200 at five energies from 2~0 GeV to 10.0 GeV. In the present experiment the elastic differential cross sections for incident proton kinetic energies of 1.0, 1,o3, and 1o.5 GeV have been measured for values of four-momentum transfer squared (-t) from 2,6 to 5~0 (GeV/c)2, corresponding to cosine of center-of-mass scattering angles 1

(cos G*) from -0o5 to -0.9. The 2~0 GeV differential cross section was measured in this experiment for -t from 0)44 to 1.54 (GeV/c)2, corresponding to cos 9* from 0.875 to 0.656. In Chapter II the backward differential cross section is interpreted as a one-nucleon exchange process, and the results are compared with the available experimental data at high energies, Chapter III considers the three-nucleon system within the framework of modern quantum mechanical three-body formalisms. The shoulderlike departure of the 2.0 GeV forward scattering data from the exponential trend of the diffraction peak is analyzed in Chapter IV via multiple scattering processes. In Chapters V through VIII the experimental techniques and results are explained, and a discussion of the conclusions is presented in Chapter IX.

II. ONE-NUCLEON EXCHANGE MODEL The one-nucleon exchange model was observed by Blankenbecler et al.l6 to peak in the backward direction. However they confined their calculation of elastic neutron-deuteron scattering to forward angles at low energies where phase shift data are available. Perl et al.17 noted that the one nucleon model could be utilized to analyze pion-nucleon elastic scattering and the reactions T- + p n+ + d and p + d + +., The one nucleon exchange diagram for p + p d + r+ has been analyzed by Heinz et al. 8 and later by Mathews et al.20 Cook et al.21 have calculated the diagram for pion-nucleon scattering but were not able to obtain good agreement with experimental data. Bernstein22 and Nearing23 have used the one nucleon exchange diagram in attempts to explain intermediate boson production via the reaction p + p + d + W+. The one neutron exchange Feynman diagram for proton-deuteron elastic scattering is given in Fig. lb with the R-matrix element. The initial proton and deuteron have four momenta Pi and dl respectively, whereas the scattered proton and deuteron have four-momenta P2 and d2. The exchanged neutron has four-momentum n. The metric is defined such that P 2 P2 = -m2 = d 2 = -M 2 p2 1 d 2 Md2 for the particles on the mass shell. The rl and r2 in the R-matrix, with r = 4r 74, are the proton-neutron-deuteron vertex functions. The relativistically invariant matrix, M, is defined by R = (2T)4 Nd Np 5 (d2 + P2 - dl - pl) M )3/2] 4 where N and Nd are the normalization factors for the proton and deuteron. The differential cross section is then da = (1) (mnd) 2 f 1 Z IM12 dn 2n U2 Pi 6 where pi and pf are the magmitudes of the initial and final three-momenta in the center-fo-mass (c.m.) system, U is the total c.m. energy and M = u(p1) r2 +m + i u(p2) T~~id t~~~~-~~ P15(;~

_ _ (a) d, o~~~n ~(b) Fig. 1. (a) Notation for center-of-mass system. (b) One-nucleon exchange Feynman diagram. The R-matrix for this diagram is: R = ( fd4n (2nrr)4 64(d2-n-pl)(2TT)4 4 3T 6 (P2+n-dl) 1 u(Pl) ~~~3 iJ3 TT ) 2TT) +m+ie 2TT r 0 21 / rT P20

The summation is taken over all proton and deuteron spin states, A. PROTON-NEUTRON-DEUTERON VERTEX FUNCTION The form of the proton-neutron-deuteron vertex function must be known to evaluate the matrix elements. The non-relativistic vertex function has been discussed by Blankenbecler et al,16 and Goldberger et alo24 A phenmenological description of the relativistic vertex function has been given by Gourdin et a1,25 The functions of Gourdin et alo were limited by the use of only low energy photodisintegration data to fix the positi ns of the three poles used to approximate the left-hand cut in the complex K plane. As a consequence, their relativistic vertex funct:ions:are.:valid:for magnitudes. of the three-momentum, K, of the neutron or proton inside the deuteron less than 0o370 GeV/c. This study of the one-nucleon-exchange diagram is concerned with large momentum components of the deuteron from 0.465 GeV/c to K = 1025 GeV/co The expression for 2 is shown to be d 2 o 1 [M 2 + (n2 + V?) + P d2+ 2 2. 2 by evaluating d2 = (n + Pl) in the rest frame of the deuteron. The vertex function of Blankenbecler, Goldberger, and Halpern16 will be considered in the following discussion, The vertex function of >Bankenbecler et alo was obtained through the use of dispersion techniques~ Although the theory was developed for all angles, they published only the real part of the forward neutron-deuteron elastic scattering amplitude at 9~6 MeV. In addition to the necessary assumptions of unitarity, time symmetry, casualty, linearity and high-energy convergence for'the derivation of dispersion relations, the following additional assumptions were made for-determination of the vertex function: (1) The deuteron is treated as a bosono (2) The discrete neutron pole is the dominant term. (3) Evaluation of the vertex function and propagator function is required only at the singularity where the intermediate neutron becomes real. (4) When the S-matrix is expressed as a retarded commutator of the proton fields, the equal-time commutator contribution vanishes identically because the deuteron cannot absorb a single pion and remain bound. (5) The number of subtractions should be no greater than the number required in nucleon-nucleon scattering. 5

(6) The discrete contributions to the absorptive part of the amplitude yield the renormalized Born approximation with the residues evaluated at the pole. When the deuteron D-State is neglected, the vertex function may be expressed by 4jtN - So m with 1 Md( ) O 2~'F M where SO is the deuteron triplet spin function, y are the Dirac matrices, N is normalization parameter, ~ is the deuteron polarization parameter, and C iZyi2 is the charge conjugation matrix with the properties C2 = 1 and CynC = y-Z~ The D-state may be included by assuming that the D-state has the same momentum dependence near the neutron poleo The deuteron triplet sping function is then expressed by S = (1 + P S12) So with S12 E ((1) r) ((2) r] /r2 (1) (2) where S12 is the usual tensor operator well known from dipole-dipole interactions aid introduced into proton-neutron interactions by Rarita and Schwinger' while p is the normalization ratio for the D-state admixture. The deuteron D-state has very little influence on the final cross sections, consequently S has been taken as the triplet spon function and the D-state 0 wave function has been represented by p times the S-state wave function. This has been evidenced by defining the normalization parameter as 2t (1 + p2) (1 -e) where II =-4 J b is the deuteron binding energy, and re is the effective range of the deuterono 6

B. VERTEX FORM FACTORS The expression for the vertex function by Blankenbecler et al. is valid only in the limit where the intermediate neutron becomes real. The vertex functions may be significantly changed as the off-the-mass-shell effects and the high momentum components are taken into account. in applying the onenucleon-exchange diagram, others 18,22,23 have found it necessary to introduce the Fourier transform of the deuteron wave function as a form factor. The form factor was used to adjust the shape and the magnitude of the cross sections to experimental data. The advantages and disadvantages of various vertex form factors will be discussed. If the vertex function can be written as r = HSo where So contains all of the spin dependent terms, Chew and Goldberger27 have shown that H = < f(r) I V(r)I e -ir > where r is the relative coordinate of the neutron and proton,, (r) is the deuteron wave function, and V(r) is the central force nuclear interaction potential. The potential V(r)' may be written as an operator V (r) = 2- _b via the SchfIdinger equation for the deuteron to show that 2 2 (a + 2) (r) e i r > m The expression for the vertex function is then 2 2 r )-(- +K) < (r) e or r =- SoF(K) m with F a2 +,2 F(K = < $(r)I e -iKor > 4tN The vertex function of Blankenbecler, Goldberger and Halpern is valid onlyin.m the limit 2 -= _2 where the neutron and the proton forming the deuteron are on the mass shell,. The value of F(K) is 1 at K = - o 28 Two deuteron:i wave functions have been discussed., by Hulthen et al. The first phenomenological wave function is V(r) = N (e- r -e -Dr),

Using this wave function, one obtains F(K) =2 K. 2 2 2 + K This form of F(K) was derived via dispersion techniques by Blankenbecler and Cook029 A second Hulthen wave function introduces an infinitely repulsive core of radius rc to jive the wave function the form r (r) = o, r < rc () = N e (r - rc r > rc r C The vertex form factor then becomes (+ K earc a, sin Krc + K cos Krc F( ) = erc.... K L 2 + K2 (a + *) sin Krc + K cos Krc (a + p)2 + K2 The constants a and N have been fixed by the binding energy and the asymptotic behavior of the wave function while the value of P, 5.18 a, is chosen to satisfy the normalization 00 4it dr r2 (1 + p2) ~2 (r) = 1. In Figo 2, F(K) is shown for several values of the core radius+ As the core radius increases, the minimum of F(K) is shifted toward smaller values of K. The Schrodinger equation for the two-nucleon potential was solved numerically by Gartenhaus using Yukawa theory with a cut-off energy for virtual mesons. The Gartenhaus wave function improves upon the Hulthen wave function for small values of ro Moravcsik31 has fitted the Gartenhaus numerical wave functions by several analytic forms, the best of which is (r) = (e'c-r e -dr) (1 - e-cr) (1 - e gr) yielding 8

;3 Iva~~~~~.2 e" rc ~~~~~~~~~~.1 *gego.I ~~~~~~~~~~~~~~~~~~~~~~~~C O"O~~~~~00 F(K) 5.6 - 7 8.9 1.0 F~ic) cosj e. e -!1 0 I es~~~f.3~~~~~~~~~~~~~~~~I F(K).'-,,, _.;! ~ 24 23 22 21 20 9 18 17 1615 14 13 12 11 10 9 K Fig. 2. Form factor from Hulthen wave function with core of various radii.

F(K) = (2 + 2) 1 1 L (2 + K2) (d2 + K2) ( 2( + )2 (2 + g)2 + K2 1 1 + 1 -+ 1 (a + + g)2 + 2 (c + d)2 + K2 + 1 1 1 (d + g)2 + 2 (c + d + g)2 + K2J where c = 6.853 c, d = 8.190 a., and g = 10.776 a. In Fig 3, F(K) is shown versus K for the three above wave functions. The form factor of the Moravcsik fit to the Gartenhaus wave function is observed to display the effects of a repulsive core similar to the phenomenological hard core in the Hulthen wave function. C. DIFFERENTIAL CROSS SECTIONS The sum of the matrix element squared given in Section A may now be explicitly obtained by substituting the proton-neutron-deuteron vertex function for r and summing over the proton and deuteron spins. The details of the spin summation are given in Appendix A, and are shown to yield 1Z M2l=,3 (2mMd- pl.d2)2 1 1 6 128 (md) 2 (n2 + m2)2 where K =2M m2 d d - n2 dld2 - mMdn (dl + d2) + 2 n.dl -d nd2 - n The differential cross section then assumethehe form da 3_2N4 d 2mMd + Pd2 X K [F ( 4 dQ 2U2m4M2 n2 + m2 j Pi 10

8$ ~1.0 GEV.?-.6.5 F(K).4.3.2.1 o.1I \I 4 8 12 16 20 24 28 32 36 K Fig. 3. Form factors for employing various wave functions. ( ) Hulthen with hard core; ( — -) Hulthen without core; (-.-.-) Moravcsik analytic fit. 11

with f = c = 1, U is the total center-of-mass energy. If the notation of Fig. la is used, the center of mass elastic differential cross section becomes 2 4 )2 72m14 d2 do 32a N 2mM + u U + k'Tcos * J d 2U2m4M2 L 2UpUd - D + 2k cos e* L02m2 + mM + mMdU2 - 2mMdUpUd + Md +MUd - 4 MUpUd 2 2 Pd d + 2UUd + X(2UU -.m..2M + k cos 9* (2k2 + 2UpUd - 3D 2md)j where k is the center-of-mass three-momentum, Up is the center-of-mass energy of the proton, and Ud is the center-of-mass energy of the deuteron. The differential cross section is displayed in Fig. 4 for different form factors, F (K). The solid curve assumes F (K) = 1, which essentially does not utilize a vertex form factor The unmodified vertex function of Blankenbecler et al. predicts a slope of the backeard peak in satisfactory agreement with the experimental data. The dashed curve shows the differential cross section with the regular Hulthen wave function determining the form factor. The slope of the backward peak becomes much steeper, and the magnitude of the peak is re duced by several orders of magnitude. The effect of using the best Moravcsik analytic fit to the Gartenhaus wave function is evident in the double-dash double-dot (-.') curve The passage of the curve through zero at bos'e* = - 0.957 is due to the form factor. In Fig. 3, the form factor obtained from the Moravcsik analytic fit is shown to pass through zero in a smooth and wellbehaved manner. The point at which it passes through zero happens to be inconvenient for application to differential cross sections below 1.5 GeV incident proton kinetic energy. The results from using the Hulth4n hard core wave function are distinguished by the dash-dot (- -.) curve. The differential cross section is observed to turn over near cos' = 1.0. The point at which the slope passes through zero is determined by the radius of the hard core as seen from Fig. 2.'To obtain maximum agreement between the available experimental data and this expression for the form factor, a radius of at least 0.71 f was necessary. The differential cross sections predicted by the onenucleon exchange diagram are shown in Figs, 5-10 along with the experimental data. The theoretical curves at all energies have been normalized to the point cos 9* = - 0.875 at 1.0 GeV incident proton kinetic energy. For each energy, the unmodified vertex function of Blankenbecler et al. yielded the best fit to the data. The form factor utilizinxg the Hulthen phenomenological hard core wave function did not fulfill his Yintended goal of 12

* TJ )TqSTU IRUs'oe (-* * —) fTDAa;noq;4T ua nflnH (- - -)'.xoo pli tr~ f U/flnH (-.-*-).T = (q), (- ) *AG@ 0'-T s SUOT40S SSOJo -sTq-ul@zJ:' p -LDTo!. OZ p@Z'T-Po fl *'T'.8 So0 01- 6- 8'- L- 9- c'- ti1000 i,,,,, s000, 100' |' _ 9001 /.. 10/'/J IIS / I qOO0'" / I I O' / I / u I:' I ~/ 09; 001 00?

35 ONE NUCLEON EXCHANGE 1.0 GEV 30 l THIS EXPERIMENT 25 BAYUKOV ET AL do d a Ser / 20 / / t', 15 / I I 5 / /~, l. I I I _I I I, I, I -4 -.5 -.6 -.7 -.8 -1.0 COS. e" Fig. 5. Theoretical cross sections at 1.0 GeV. F() = 1;( Hulthen with hard core; ( — -) Hulthen without core. 14

24 22 20 1.3 GEV 0 THIS EXPERIMENT ONE NUCLEON EXCHANGE 1s do Fig. 6.12 Theoretical'cross-sections at 1.3 GeV. ( ) F( = 1; (-. l ~.ulthen with hard core. ~ ~ ~ ~ ~ ~ ~ t1 12 I0 eP4~4 n~~./ - a~2 -.4 -5 -.6 -.7 -.8 -.9 -1.0 cos e* Fig. 6. Theoretical cross sections at 1.3 GeV. (F(K); Hulthen with hard core. 15

20 18 16 14 de 12 da 1.5 GEV o THIS EXPERIMENT 10 stL — ONE NUCLEON EXCHANGE 6 / _ 4 iCOS V cos d' Fig. 7. Theoretical cross sections at 1.5 GeV. ( ) () = 1; (Hulthen with hard core. 16

100 0.660 GeV * LE(SIM ONE NUCLEON EXCHANGE 0 70 30 20 10 Fig. 8. Theoreticai cross se t o a at O 6 G V (- - ) (1) =; Hu tihti with hard Core. Sections. 0.660 Gek -- 0 le

60 0.715 GEV * BAYUKOV ET AL 50 40 dcr dSI 30 Ster 20 10 -4 -5 6 -,8 -.9 -1.0 Fig. 9. Theoretical cross sections at 0.715 GeV. ( ) F(G) ( —-.-) Hulthen with hard core.

,I' I'I', 10 9 8 3.66 GEV * BAYUKOV ET AL 7 6 do 5 pib Ste r. 4 3 2 I _.6 -7 -.8 -.9 -1.0 COS. 8' Fig. 10. Theoretical cross section at 3.66 GeV. ( ) F() (-.-.-) Hulthen with hard core. 19

accurately representing the high-momentum components of the deuteron at the energies studied here. The regular Hulthe/n wave function and the Moravcisk fit to the Gartenhaus wave function suggested form factors which show poor agreement with the data. The Moravcsik form factor may, however, show better agreement with the data than the Hulthen hard core form factor at higher energies because before the catastrophic influence of passing through zero the shape of the predicted differential cross section was in good agreement with the experimental slope. 20

III. THE THREE-NUCLEON SYSTEM The quantum mechanical three-body scattering problem has recently become tractable through new formalisms. -The old methods used various perturbative operations which introduced many phenomenological three-body parameters to fit experimental data and tended to overlook the requirement that the bound states and the scattering states be dictated by the same dynamical relationships. Unlike the two-body problem, a choice of coordinate frames cannot reduce the apparent number of particles in the three-body problem. The six independent momenta in the center-of-mass system and angular momentum considerations complicate the kinematical relationships. The multichannel process for quantum mechanical systems was put on a rigorous mathematical foundation by Jauch.34 He derived explicit integral representations of the wave operators. However, the appearance of delta functions in the kernel of the LippmanSchwinger equation 35 made classical iteration methods impractical. Also the homogeneous Lippman-Schwinger equation possessed solutions when bound states existed between pairs of particles which made the problem of satisfying boundary conditions quite difficult. In three-body processes it is possible for two particles to interact while the third particle does not interact. This leads to disconnected graphs which yield integral equations with singular kernels. These equations cannot be solved by the Fredholm method, perturbative methods or any other known form of computation. The three-body problem was cast into a form solvable through present analytical means by Faddeev.36 A set of three-body equations was given in which the kernel had no continuous spectrum and for which the hgmogeneous equation possessed a solution only for bound states of the entire system. The kernels are generalizations of the T matrix and may be determined by working only with pair amplitudes. The coupled integral equations describe all possible processes of the three-particle system without the introduction of free three-body parameters, and the three-body amplitudes obey unitarity on and off the energy shell. The total angular momentum has been separated in the Faddeev equations by Omnes37 in a manner which preserves the symmetry of the problem with respect to the three particles. The many degrees of freedom in the Faddeev equations make them difficult to solve numerically. This may be resolved by approximating the two-body potential as a sum of separable potentials. Mitra38 has shown that exact three-body equations are obtained involving only a few coordinates in the intermediate states if the two-body operator is compact. If the parameter of the separable potentials is adjusted suitably, the two-body bound state may be considered in the calculation. The separable potential method is identical in the limit of zero wave function renormalization for the two-body bound state with the quasi-particle method of Amado39 and Weinberg.40 This calculation will follow the field theoretic method of Amado because of the p~hysical clarity of his presentation. 21

The idea of substituting elementary particles for composite systems was rigorously studied by Jauch34 and by Vaughn, Aaron, and Amado.41 Jauch pointed out that the distinction.. between elementary particles and composite fragments is a superficial one from the view of scattering theory. The picture of the composite particle as an elementary particle only breaks down when the internal structure reveals itself in breakup processes. Vaughn, Aaron, and Amado showed that the nonadiabatic behavior of the composite system is overcome by introducing it originally as an elementary particle. Then soluable linear integral equations are derived, whose solutions exactly represent scattering, breakup or stripping reactions. Amado39 later displayed how these equations could be put into the form of the Lippman-Schwinger equations. Since linear integral equations are derived, off-the-energy shell amplitudes are involved if threebody unitarity is to be satisfied. Remaining on the energy shell requires the solution of nonlinear integral equations. In the following the incident proton will be on the energy shell, and naturally the outgoing proton will be off the energy shell to define the integral equations. Aside from elastic scattering, deuteron breakup into a proton and a neutron will be the only reaction permitted in the following paragraphs. In the energy region considered by this work, deuteron breakup is overwhelmingly the major contribution to the total cross section. Using the notation of Aaron, Amado, and Yam42 the spintriplet pair interacts to form a "d" and the spin-singlet pair interacts to form a "p". The spin-triplet and spin-singlet scattering amplitudes are assumed to have no coupling between them, and they are:given nonrelativistically by < k d t (E) I k' d > = Xdd < k d B(E) I', d > (2 1 Xd a <f d | B(E) d > P (p;E) <,dl t(E) I?',d:> 1 k, E) < (23 d P Xdp < k B(E) p > p (p2; E), c t(E),d > + 1, d3,~p X~d<'- B(E) |p,d > P p2;E) < p j t(E) I',d > (2~)3 + 1 fp Xd,p B (E) t p t(E) I >tk',d > (2 )3 22

where't = 2m = 1, B(E) is the Born approximation for the exchange of a nucleon between pairs, P(p2;E) is the full propagator in the intermediate state, and the X's are the spin and isospin factors. The graphical representation of these equations is given in Fig. 11. The explicit form of the Born terms in the integral equations are < %,4 I B(E) I %',d > = 2 Vd + i / ) ]d it22/2] + < kp J B(E) I k > = r2 Vm [it + /2) ] Vq [ + /2)27 E k2 k?2 _-(k + k)2+is =cPd E. k2. k'2+ =~ rd,2.1-'k2 - k,2 - (k + 2') + ic where rd and r. are the coupling constants to the "d" and "Tcp quasiparticles, Vd and Vet are the vertex form factors. The propagators for the intermediate states have the form F 2 V (n2) V (n2) Pd (P2;E) = ( + b)fdn d d _L(2E ((.2ir)3 (2n2 + b)2 ( - 2n + i P (p f;E) = - L + f fd3n ) (2j)3 (c - 2n + ic) where F = E 31p2 and b is the binding energy of the deuteron. The introduction of separable potentials is evidenced by the appearance of only one vector variable, p, labeling the intermediate states. The form of the vertex form factors must be chosen so that the kernel is square integrable in order to use the Fredholm method of solving these equations. Amado has proven that this condition is satisfied in momentum space by V(K2) = c/l +, ~ > o for large X which corresponds to investigating the potential at small distances in configuration space. Using the momentum representation of the Hulthe/n wave 23

' d 7 k Fig. 11. Diagrams for proton-deuteron scattering. The nucleon is a single line; the double lines are p for singlet and d for triplet; the small circles are nucleon-nucleon vertices; the rectangles and large circles are three-nucleon amplitudes. 24

function yields Vd(,2i = 1 and V(K2)= 1 d2 + d2 + p2 which approximates the left-hand cut in the complex three momentum plane by a pole at K2 = _ 2o Only two-body nucleon-nucleon parameters enter the calculations, and in terms of the triplet scattering length at, the singlet scattering length as and the singlet effective range rs they are given by 1 ad Ed (2 d +) at 2 (ad + Pd)2 d= 32 Tc Ad Pd (ad + adz3 r2 2 = 16 [ as cP as ABt 2 3 11 rs 1~3r F]+(lr s) I] P =2rs 1 9 as j where =Ad b/2. The integral equations may be solved by the Fredholm inversion method435 for low energies, but at the energies considered here Yam44 has shown that the Neumann series for the equations converges. The first term of the expansion is the Born approximation. The unnormalized differential cross section predicted by the Born term is plotted in Fig. 12 both with the Hulthen form factor and with the form factor set equal to one, (i.e.,V (K2) = 1). The spin average of the differential cross section when only the Born term is included takes the form dcr 2 2 2 d=l 3 | Xdd k d B(E) | knd) + ( B(E) | k'd) The differential cross sections normalized to the point cos G* = -.875 at 1.0 GeV incident proton kinetic energy are shown in Figs. 13-16 For the first term of the iteration the differential cross section has a slope which is much too steep when the form factor is included. The use of other deuteron wave functions in determining the form factors (see Chapter II) does not improve matters significantly. Further iterations of the equations including the form factor only increases the slope of the backward peak as the amplitude 25

9Z aO4DosJ UIQJ UGq4rTlnH (-. —)'I = (a )A ( )'I*A 0T' 1 s suan uLoq paz TTELouun' T'J 8 08 Soo 01- 6- 8- L- 9'- 9- - 100, I O 0' / 10 0''' 1.,' / qw/ / UP, 01. /:O~ddU NEON g 0./ Up~~ j.op $!. 01 ~ ddV dO qW~ ~ A90 0I~ 00'

35 1.0 GEV i 30 BORN APPROX. l oTHIS EXPERIMENT * BAYUKOV ET AL Str. 15,.,I -10 ImI -4 6 -5 -.6 -.7 -.8 -.9 -1.0 Fig. 13. Contribution of born terms at 1.0 GeV. ( ) V(K2) 1 (-*-*-) Hulthen form factor. 27 -5 Fig. 2K5. Cotribution ofborn terms t 1.0 GeV. ( ) V(~ ) = I (-.-.-) HuZ~~~~then form factor.~ ~~ 27~~~~~~~~~~~

25 20 1.3 GEV BORN APPROX. I oTHIS EXPERIMENT 15 do Str.; IC I0 -.4 -5 -.6.7 8 -.9 -I.0.4 -5 -.6 -.7 -8 -9 -1.0 COS 8a Fig. 14. Contribution of born terms at 1.3 GeV. ( ) V( 2) = 1; (-.-.-) Hulthen form factor. 28

15 U'' ~''' W'' ~ I''' W U I' I. 1.5 GEV BORN APPROX. oTHIS EXPERIMENT I0 doStr. i I i i 5 i / / -.4 -5 -.6 -7 -.8 -9 -1.0 COS e' Fig. 15. Contribution of born terms at 1.5 GeV. ( ) V( 2) = 1; (-.-.-) Hulthen form factor. 29

IWI ~ II' -70 0.660 GEV 65! BORN APPROX 60 LEKSIN I 55 i i -i I /50 do - 45 d 40Str. 35 i I r'~30 i 25 / /20 -~~~~~~~~~~~~~~~ I - 3. ~~~~~~~~~~/ 15 10 5.4 -.5 -.6 -.8 -.9 -1.0 cos o' (-~ —) Hulth'n form factor. / ~

raises in the forward direction to satisfy unitarity. Consequently when the vertex form factors are included, the experimental data is not adequately predicted because the high momentum components of the deuteron are suppressed too greatly. When the form factor equals one, the kernel of the integral equations is not square integrable as mentioned above, and the Fredholm method is not applicable. - The Neumann expansion in this calculation depends upon the approximation of the potential by a finite sum of separable potentials, and the factors of the separable potentials are proportional to the bound state form factors.45 In the three-body system the Neumann expansion differs from the Born expansion because in general the kernel is not a linear function of the two-body coupling constant. When the Hulthen form factors are used, the Neumann series converges for sufficiently high energies for any partial wave. However, when the form factors are set equal to one, the convergence of the Neumann series is not readily proven because the kernel is not known to be compact. The Born series diverges independently of the incident energy of the proton whenever a twobody bound state enters the three-body system.46 The zeroth iteration, which is the Born approximation, is in closest agreement with the experimental data. The calculations then reduce to the one-nucleon exchange diagram with a nonrelativistic propagator In the discussion of Chapter IX, the results of this chapter and Chapter II will be compared. 31

IVo OFF THE FORWARD DIFFRACTION PEAK Elastic differential cross sections for scattering of elementary particles at high energies generally have a large peak in the forward direction which has been denoted'l ithe diffraction peak. The diffraction peakThe diffraction peak has been studiedfor proton-deuteron elastic scattering in the few GeV region by Kirillova et al.,but the analysis was not carried beyond this peak due to the lack of data. The present experiment observed a shoulderlike departure from the strict exponential trend of the differential cross section for four-momentum transfer greater than 0.5 (GeV/c)2 at 2.0 GeV incident proton kinetic energy.48:Previously, secondary peaks or shoulderlike structures have been noticed in pionproton elastic scattering, kaon-proton elastic scattering, pion-proton charge exchange scattering, kaon-proton charge exchange scattering, and positive pionrho elastic scattering.49- 55 However the secondary peak does not appear in proton-proton and proton-neutron elastic scattering.56 No satisfactory theoretical model for the secondary diffraction peak has been advanced. The most common method of studying the secondary peak has been via the optical model. As pointed out by Simmons,57 the position, height, and to some degree the width of tht secondary diffraction peak in pion-nucleon scattering may be fitted with the.optical model. However, the differential cross section goes to zero as many times as the value of the maximum angular momentum con-.' sideredo For proton-deuteron elastic scattering at 2.0 GeV incident proton kinetic energy, the model would predict a differential cross section which goes to zero fourteen times since Lmaxy kR- 14. If a diffuse edge is added to the deuteron by assuming small contributions to the differential cross section by values of angular momentum greater than Lmax, the postion and magnitude of the secondary peak are adversely altered although the number of zeroes may be reduced. Due to the composite nature of the deuteron, other avenues of approach are available which are ostensibly closed when the internal structure of the target particle is unknown. The following methods make htse of the known composite structure of the deuteron in an attempt to interpret the behavior of the proton-deuteron elastic differential cross section off the forward diffraction peak. A. IMPULSE APPROXIMATION The standard impulse approximation, which neglected multiple scattering effects, gave a reasonable description of proton-deuteron elastic scattering at small angles in the few MeV region.58,59 The basic assumptions were that the incident proton never interacted strongly with the two components of the deuteron at the same time, and the binding force of the deuteron was negligible during the interaction time. If the incident proton is number 1 and the target proton and neutron are numbers 2 and 3 respectively, let V2 and V3 re32

present the two-body interaction of the incident proton with the proton and neutron. Let U be the interaction between the constituents of the deuteron, E be the total energy of the system, and K be the total kinetic energy operator of the system. The total Hamiltonian for the system is then H = K + U + V2 + V3 Ho +V where Ho = K + U is the unperturbed Hamiltonian. The scattering matrix, T, may be written, as specified by Chew and Goldberger,60 in the form T = V + VGV with G = lim E - H + ie Then neglecting multiple scattering, T may be written as T = St2 + St + 2 1 where 1 where t2 V2 2 E - K - V + iE V2 t3 V3 + V3 E - K - V3 + i V3 and S is the "sticking factor" defined by Chew63 to be 4.. iq. r S(q/2) = fdr e where q is the three-momentum transfer and V (r) is the deuteron wave function. In the two-body problem, the relationship between q and the center-of-mass angle G * is q = kL sin (02*/2) where kL is the momentum of the incident proton when the target proton or neutron is at rest. For the three-body problem the three-momentum transfer remains the same, but the relationship to the three-body center-of-mass scattering angle e~ is then q = (4/3) kLsin 6e/2. Consequently the scattering angle of the proton-deuteron system is related to 33

the two nucleon scattering by e# 4 f4 sin sin 2 3 2 which yields the center-of-mass solid angle transformation dQ2 16 4~23 9 dq2J ~~~~~~lChew63 in terms of momentum transfer aspin where fnp and fpp are the spin non-flip amplitudes and, f are the spin flip amplitudes. The relative )phase of the proton-proton and neutron-proton amplitudes and the difference in magnitudes of the spin flip and spin nonflip amplitudes are not known at the energies of concern in this work. Neglecting the D-state of the deuteron and only considering the spin dependence required by the Pauli exclusion principle yielded the differential cross section in the form dpd - 16 5danp 3 d pp + dnp d 1 pps\ ) 7o~d + - d +_ dn2 Cos A ( dQ3 9 dQ2 4 dQ2 \dQ2 dQ2 in the center-of-mass system where A is the difference between the neutronproton and proton-proton phase shifts. The S (es) or the corresponding S(q/2) was determined by using the regular Hulthen wave function and the best Moravcsik fit to the Gartenhaus wave function (see Chapter II, Section B)e Using the Hulthen wave function yielded the relationship S(q/2) = 8 tan-1 L + tan-1 (a'' hereas the Moto the Gatehas wave ctio ielded 34

S(q/2) _ r an -F + -tan, -- + tan- l q 4a 48 4"a + 4c + tan-1 q + tan-1 q +tann- 4a + 4g 4 (a + c + g) 4c + 4d + tan — q + tan-1 -2 tan'-l 4d + 4g 4 (c + d + g) 2c + 2d -2 tan-1 q -2 tan-1 q +4 tan'1 q 4ce + 2c 4A + 2g 4a + 2c + 2g + 4 tan-1 q + 4 -8tan-1 q 2 (c + c + d) 2 (c + d + g) 2 (a + c + d + g) -2 tan-l ~ -2 tan-1 + 4 tanl 2 (c + 2d) 2 (2d + g) 2 (c + 2d + g) -2 tan-1 q -2 tan-1 4 +44 t an-1 2 (2 + 2c + g) 2 (a + 2c + d) 2 (c + 2c -+ d + g) -2 tan-1. -2 tan-1 q +-4 tan-,1 2 (2c + c + 2g) 2 ( + d +g) 2 (a + c + d +2g) -2 tantan c + 2 d - 1 qq 2(c~+2c~d+2g)-2 tan- 1 2 2 2 (2~+ c + d2g) 2 (a 2c + d (+ 2c + + 2 g) Due *to the lack of suitable neutron-proton phase shift data in this region, the phase shift difference has been set equal to zero which implies that the neutron-proton and proton-proton differential cross sections are equal. The predicted proton-deuteron elastic differential cross section at 2.0 GeV incident proton kinetic energy is shown in Fig. 17 using the experimental proton-proton elastic scattering data 6f Barge64 and the S(q/2) from the Hulthe'n wave function. The use of the Moravcsik fit to the Gartenhaus wave function gives essentially the same results. The impusle approximation is in good agreement with the small angle data of Kirillova et al.,15 but the strict exponential trend of the curve does not reproduce the shoulderlike departure of our data from the diffraction peak. Since there is no evidence of a secondary maximum in either proton-proton or neutron-proton elastic scattering, there is no reason to expect the impulse approximation to predict one in proton-deuteron scattering. However if multiple scattering is allowed, the theoretical curve

I10,000 - 2.0 GEV 5 THIS EXPERIMENT * KIRILLOVA ET AL ~ OPTICAL POINT ---- IMPULSE APPROX. 1000 500 dod/S.ub Str 100 50 -.2 4.6 8 1.0 12 14 16 1.8 20 -t (GEV/C)2 Fig. 17. Prediction of impulse approximation at 2,0 GeV. Differential cross section in center-of-mass system. 36

may tend to follow the experimental data more closely. B. MULTIPLE SCATTERING PROCESSES The effect of binding and multiple scattering processes may be studied as corrections to the impulse approximation as has been shown by Everett.65 For small momentum transfers, Chew and Wick62 have estimated that single scattering dominates when the condition (-2 k 1 -R7 4ic is satisfied where R is the average internucleon distance, X is the wavelength of the incident proton divided by 2~i, and a is the total cross section. At 2.0 GeV the left side' of'_the inequality equals.0,049, and as seen ini'Section.:A the single scattering processes indeed dominate on the diffraction peak. However, for larger momentum transfers around -1.0 (GeV/c)2 the inclusion of multiple scattering may have a more noticeable -effect than expected from the above inequality. An analysis of multiple scattering in the collision of particles with deuterons using a generalized form of diffraction theory has been advanced by Franco and Glauber.66 This work follows the method of Franco and Glauber because of the simple physical interpretation of the scattering effects as they appear in the terms of differential cross sections. Near the forward direction, the scattering amplitude at high energies may be written as f(kg)- ik fexp i (t - %') 1 - exp (iX (u| du where u is the impact-parameter perpendicular to the direction of the beam and X (t) is the phase shift. For a target particle consisting of a total of A nucleons with positions rl,..,rA, the wave of the incident particle will undergo a total phase shift Xtot (,l.....A). If the incident particle interacted only via two-body interactions with the target nucleons, the total phase shift would be the sum of the individual phase shifts. If s is the projection of the postibn coordinates on a plane perpendicular to the incident beam the total phase shift may be expressed by A Xtot (u) rA X xjr ( -A I5 37

For the deuteron, let r and rp be the coordinates of the neutron and proton n P prespectively. Then the relative internal coordinate is p = Tp - tn. The total phase shift is Xtot = Xn (u - ) +Xp ( + ) with s now the projection of r. Let the function rtot (t, 1) = 1 - exp [EXtot ( be introduced for abbreviation purposes. Since the comparison of the protondeuteron scattering amplitude with the proton-proton and neutron-proton scattering amplitudes would require two center-of-mass systems, the following calculations refer to the laboratory system for convenience in notation. The scattering amplitude is then the matrix element between the initial state ii > to the final state If>, which is Ffi (j'j)ik e\ ~2Ffi (k )= 2< e exp (-') ~ u < f I rtot ( u, s) I i > du the contributions from single scattering and double scattering are considered here, and the appropriate diagram of the calculated processes is shown in Fig. 18. Tertiary and higher order scattering processes involve at least one or more backward scatterings and have a negligible effect on forward protondeuteron elastic scattering. If we let the momentum transfer be represented by - = h (~ - $) and let the energy transfer to the target particle be negligible for forward scattering, the elastic scattering amplitude assumes the form Fii (i ) ik2 exp (i. u) de f r (r) rtot (U, )'i (P) d The scattering contributions of the individual nucleons may be separated by defining rn = 1 - exp iXn (uTh and r =1 - exp ( n p ~ ~i3 to obtain the identity rtot = (u ) + rp ( + -) F 1 + ) 2 2 Then substituting this expression in the above equation and shifting the origin in the u plane yields Fi + ( i) ik [ exp ( iqs) i (r)dfrexp (i.7U) Fn (U) dU 38

Pp P P P dd.d P P P P P P + p +n d " d dP d- Fig. 18. Diagrams for single and double scattering. The impulse approximation only considers the first two graphs on the right hand side. The influence of all four graphs on the differential cross section is considered in the calculations of Section B. 39

+ 2 f exp( 2 ) Vi (r) dr S exp (iq u) rp (u) du _ik fi (r)' dr f exp (iq U)n( - s) r! (u + 2Tc P, 2 Using the expression for the sticking factor defined in Section A and noting that P(u) is the Fourier transform of f(q), that is (u) =- -- exp (-q.u) f (q) dq the elastic scattering amplitude takes the form Fii (q) = s(-') fp (q)S( fn s( _ i! ->~ -t L? 9'*'* 4 +.2k f S(') fn (.2 + g') d The elastic differential cross section is the square of Fii (q) and may be written as d = S2 21) f +lfn + 2 Re f )f - q-1 n-)I | (q)|n (R )f -~ i: ~) Im l f~ (~)+ f* ( ) I S ( + q') f q -Th dqciv ~.'_1 2.(2..kJ n (""~ *') fp ( - 8') d2' | The contributions from single and double scattering appear explicitly in the above equation. The first term corresponds to the scattering by a free proton and by a free neutron and the interference of the two wave amplitudes respectively. The.second term corresponds to the interference between the doublescattering amplitude and the neutron single-scattering amplitude plus the interference between the double-scattering amplitude and:the proton single-scattering amplitude. The third term corresponds to solely double-scattering. 40

For application of the differential cross section formula, expressions must be found. for the scattering amplitudes fp (_4) and fn(d). The amplitudes have been assumed to be of the Gaussian form in the momentum transfer and are represented by fp (4) = (i + Cp) (kap/4t) exp (-Apc2/2) fn () = (i + An) (kan/4it) exp (-Anq2/2) where ap is total proton-proton cross section, an is total proton-neutron cross sectioni, cp is the ratio of the real tob-the imaginary part of the proton-proton scattering amplitude, an is the ratio of the real to the imaginary part of the proton-neutron scattering amplitude, A is the slope of the proton-proton elastic differential cross section, and An is the slope' of the proton-neutron elastic differential cross section. Substituting thes'd expressions for the amplitudes into the differential cross section yields d~~2 n n p p9)2 + 2(1 + Ctrpp) anp e l(An + Ap) 2 n p (A + ) ne/ n 2p (1 + ip L ( + 2)an e 2(An p ApA2 - An)~O4, - i(A + Ap)2 S(>') e -AP(Ap e 2 n p d' 2 2 2 _ 2 S() e 2(Ap- An) q.q' e - (An + A)q 2 de' 2 Although the cp and ocn may not be equal as reported by Kirillova et al. 15 the slopes of the differential corss sections have about the same values as recently measured by Kreisler et al.56 Letting An = Ap = A simplifies the e.quation to 41

d 2 (i q) (k )e -Aq2 F(1l + C 2 + (l+ )a2 + + ( + + 2 (1 + n d-~ 2 ( q ( en n p npnp 22 ] k2anP e -Aq2/4 [1+ + + n(4 4 + 2 JS(q )e dq2 2 + ( + a +. + 4+c) e - fS (q')e A dq'12 It is interesting to note the magnitude of the error in the An determined by Kirillova et al., who found %n = 0.2 + 0.4 and czp = -0.12 + 0.07. The values of An and up determine the relative phase between the single and double scattering amplitudes, and the effect of varying cn to the minimum experimental value is shown in Fig. 19. A theoretical evaluation of the real part of the proton-proton forward scattering amplitude has been published by Soding.82 By using dispersion relations with the integration in the unphysical region replaced by the p meson and X meson poles, S'5ding obtained ap = -0.27 for 2.0 GeV incident proton kinetic energy. The real part of the proton-neutron forward scattering amplitude has been determined by Carter and Bugg, who fitted the low energy contribution from the unphysical region with only the p pole.'3 Carter and Bugg found An = -0.50 at 2.0 GeV. The dip in the differential cross section is much less pronounced when the an and ap assume the more negative values of the theoretical predictions as evidenced in Fig. 19. Using the S(q/2) determined by the Moravcsik analytic fit to the Gartenhaus wave function and the predicted values of cn and 4p, the theoretical differential cross section is displayed in Fig. 20. The theoretical curve is in good agreement with the experimental data in slope and magnitude except at the dip caused by the interference terms, There are no free parameters in the formulation since all quantities are set br nucleon-nucleon data. The valley at It| = 0.38 (GeV/c) has no apparent justification, but perhaps it can be eliminated by selecting amplitudes which represent the scattering at higher momentum transfers better than the Gaussian amplitude. 42

50soot- 2.0 GeV an=-0.5, ap -0.27 100daZ d 50 /,Fb str = I.\ t/ian- 0.2 1ap=-0.12 5 1 \-a -0.2, ap =-0.12 I0.3.35 4 45.5.55 Itl Fig. 19. Effect of the real parts of the nucleon-nucleon scattering amplitudes on the proton-deuteron differential cross section.

500000ooooo 20 GeV * THIS EXPERIMENT A KIRILLOVA ET AL. 00000oooo -- INCLUDING DOUBLE 500 oooo, SCATTERING 50000 10000 d o 5000 st r 1000 500 - )00 -~0 100 50-.5.2 4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 I tI Fig. 20. Theoretical differential cross section in the laboratory system when single- and double-scattering interactions are included. 44

V. EXPERIMENTAL PROCEDURES The differential cross sections were measured for backward proton-deuteron elastic scattering at 1.0, 1.3, and 1.5 GeV incident proton kinetic energies and for forward scattering at 2.0 GeV. Protons at the desired energies were obtained from the external beam 2A of the Cosmotron at Brookhaven National Laboratory. The beam was analyzed by a system of three bending magnets and three quadrupoles and focused on a three-inch-long liquid deuterium target. The primary difficulty in determining the differential cross sections at high momentum transfers is that the magnitude of the high momentum transfer portions of the differential cross section is several thousand times smaller than the forward peak. The background from the deuteron breakup processes, which yield many protons available to trigger the counters for each deuteron of elastic scattering processes, is quite large at the energies considered in this experiment. The problem of the immense background was overcome by detecting both of the scattered particles instead of only one as is the case of most proton-deuteron elastic scattering experiments and in addition by performing momentum separation and time-of-flight analysis on one of the scattered particles. By this procedure the chance events were effectively reduced to less than 10% of the desired events. A. BEAM The beam was pulsed at 30 pulses per minute yielding 8 x 107 to 1 x jo9 protons per pulse for convenient rates of accidentals. The kinetic energy of the protons was determined by two methods. The first method was to measure the frequency, f, and orbital path length, L, of the internal beam. If the proton rest mass equals M, the kinetic energy is known to be T = M[ -Lf)2 where c = 1. The second method consisted in measuring the Cosmotron magnetic field, B, and the radius of the circulating beam, R, which determine the kinetic energy through the relationship T = (M2 + e2B2R2)l/2 _ M where e is the proton charge.

Bennett has found the Cosmotron to have an inherent energy spread of 2 MeV,67 The two deuteron telescopes also provided a check on the beam energy (see Section D). These methods determined the beam energy to within 1%. B. BEAM OPTICS The components of the beam optical system are shown in Fig. 21. The currents in the components H200 to H205 were determined by a beam-design computer program to first order, The fine tuning of the magnets were empirically made by exposing Polaroid film simultaneously at the target position and a beam line counter position. The final current values with their energy dependence is given in Table I. The bending magnet H200 served two purposes. The angle at which the beam leaves the Cosmotron ring depends upon the beam energy. H200 allowed one to compensate for the energy dependence of the exit angle by regulating its magnetic field. A collimator was placed in the gap of H200 to provide additional control of the beam spot size. As seen in Table II, the pole faces restricted the spot size vertically to 1-1/2 in. The collimator allowed one to simultaneously satisfy conditions of small spot size and small angular divergence for the beam. Three quadrupoles focused the beam at the target. Q201 and Q203 focused the beam horizontally'while Q202 provided vertical focusing. The currents in these quadrupoles were adjusted to restrict the beam to a maximum angular divergence of 3-1/2 mrad and a maximum spot size of 1-1/2 in, at the target. The bending magnets H204 and H205 guided the beam along the axis defined by the center of the target and the center of the C counter. Bending magnet H206 (Fig. 22) provided momentum analysis of the scattered deuterons at incident proton kinetic energies of 1.0, 1,3, and 1.5 GeV and of the scattered proton at 2.0 GeV. This magnet will be discussed with greater detail in Section D. C. LIQUID DEUTERIUM TARGET The proton beam passed through at.010 ino mylar window to reach the vacuum chamber enclosing the target. The target was connected by a gravity fill line to an overhead reservoir, which maintained liquid deuterium at a temperature of 207~0 + 0.2~Ko Liquid deuterium at this temperature has a density of.1697 +.0002 g/cc. A.010 in. mylar cylinder with mylar domes on the end constituted the target. The cylinder was 3.00 in. in diameter and with the domes had an overall length of 3.00 in. in the median plane. For thermal insulation, the target cylinder was wrapped with 20 layers of.00025 in. NRC Super Insulation (aluminzed mylar) in addition to evacuating the surrounding chamber. The reservoir was wrapped with 120 layers of the NRC insulation. The scattered particles passed through a,014 in. U-shaped window in emerging from the vacuum chamber,

C RON 3"~~~~~~~~~~~~~~~~~~~~~~~3 LIO. Di TARGETT H 200 SCALE (Irrrl~r7lot I~~49~~Fg.2. hebamlyot

TABLE I MAGNET CURRENTS (AMPS) Magnet Tp 1.0 GeV 1.3 GeV 1.5 GeV 2.0 GeV H200 70 55 50 28 Q201 116 137 146 200 Q202 290 335 365 477 Q203 240 275 300 375 H204 168 235 244 364 H205 745 846 989 127? TABLE II MAGNETIC CHARACTERISTICS Magnet Size Max. Current Max. Mag. Flux (amps) Density (k Gauss) H200 1.5"x6T"x12" 250 11.4 Q201 81'x16" 1000 11.5 Q202 8"x32" 1000 11.5 Q203 18"1x32" 1000 11.5 H204 18"x36" 800 22.7 H205 12 "x60" 1820 17.6 48

Ds D? A rLIQ.Dit OBB EXTERNAA L TA ME T o 1 0 ~~D PlhUM ID COSMOTR sGALEe P~~~~~~ I 0 5FT SMIELDING Fig. 22. Plan view of experiment.

Do DETERMINATION OF SCATTERING ANGLES The kinematical relationships for proton-deuteron elastic scattering were determined by an IBM 7090 computing program to less than.01%OL uncertainty. The center of the target and the center of the pivot for detection telescopes was determined; then a zero degree beam line was constructed. A theodolite was centered on top of the pivot post to determine the polar angles of both scattered particles. A second theodolite was used to record the height of the center of the target at convenient locations on the shielding walls. These reference points were then utilized to position the centers of the scintillation counters at beam height. The first counter of the P telescope (Fig. 22) determined the solid angle of acceptance. In backward scattering of the incident proton, the shorter arm determined the polar angle of the scattered proton. The angle between this telescope and the beam line was varied by rotating the I-beam, which was connected to the pivot under the theodolite and rolled on a leveled steel platform. The longer arm determined the polar angle of the scattered deuteron in foward scattering of the incident proton collisions. This arm was varied by rolling the I-beam along a circular rail located 19 ft from the target. The P1 and P2 were aligned along the center of the I-beam using the theodolite mounted over the pivot post. The center of the P1 was set with the theodolite to the polar-angle specified by the kinematic tables mentioned above. The deuteron (proton) in backward (forward) scattering of the incident proton was detected by two overlapping telescopes consisting of three counters each, These counters constituted the D telescopes and were referred to as D1A, D2A, D3A, and DlB, D2B, D3B. The Dl and D2 counters were supported by a motorized carriage, which also supported a bending magnet H206 between them, H206 deflected the particles passing through D1 by +100 to provide momentum separation of the desired elastically scattered particles from quasi-elastic events and reaction products. The 18 in. x 36 in. H206 attained a maximum current of 1000 amps, which was equivalent to a magnetic flux density of 15.3 k gauss with a 10-1/2 in, gap. This was sufficient to separate the desired particle up to 2.4 GeV/c momentum. The relationship between the current in H206 and the cosine of the center-of-mass scattering angle of the incident proton (cos 9*) is given in Fig. 23. In addition to H206, the DA and DB channels further increased the momentum resolution by requiring symmetry between the channels. The 40-foot flight path between the D1 and D3 counters provided a time-of-flight criteron which the scattered particle was required to satisfy. The angle of the D1 counters was conveniently adjusted by moving the motorized carriage until the center of the overlapping portion was aligned with the vertical cross hair of the theodolite. The overlap of the D telescopes was determined from multiple Coulomb scattering in the target, and, and any previous counters. The angle, momentum selection, and time-of-flight restriction of the D channel together with the angle of the P channel provided a kinematical check upon the incident beam energy. The angle limitations of the differential cross sections measured in the backward direction were dic5o

COS 8' FOR 2.0 GEV.4.5.6.7.8.9 1.0 1000,- 2.0 GEV 800 -' 1.5 GEV 1.3 GEV 1.0 GEV 400 200 1.0OGEV 1.3 GEV - 600 - GEV - -2.0GEV -800 -1000 -4 -.5 -6 -.7 -.8 -.9 -1I.0COS 8' FOR L0, 1.3, 1.5 GE V Fig. 23. Current in bending magnet H206 vs. cos 9* at various energies. 51

tated by the positions of the shielding walls in the experimental area. In the laboratory system, the Dl counter was limited to a maximum angle of 280 from the beam line and the P1 counter was limited to a rMaximum angle of 1220 from the beam line. The forward differential cross section measurements were limited by the requirement that both scattered particles reach the final counters within the 8 ns time resolution (see Section E) and with less than 10% accidental event counting statistics. At very forward angles the energy loss of the recoil deuteron during traversal of the scintillation counters was too great to satisfy the above criteria. E. DETECTION OF EVENTS The elastically scattered protons and deuterons were detected by scintillation counters using fast electronic circuitry. The positions of the counters are displayed in Fig. 22. The scintillation material of the counters had a polystyrene base and was commercially available. Except for the M monitor counters, the light from the scintillator was transported to RCA 6810 photomultiplier tubes by Plexiglas light pipes. The scintillator of the M monitor counters was connected directly to the photomultiplier tubes with R-313 epoxy to decrease the small inefficiency introduced by light pipes. The size of the scintillator for each counter in shown in Table III. Each of the counters was checked by a beta source before and after the experiment to ascertain whether the counters remained uniformly sensitive over the area of the scintillator.- Several times during the experiment, the counters were thoroughly examined for leaks. The counter supply voltage versus counting rate showed a plateau, and each counter was operated on its plateau, while determining the time delay between incidence of a particle and its output pulse using a light pulser. The D3 counters had a photomultiplier tube on each end of the scintillator to reduce the signal to noise ratio and to decrease the characteristic timing variation caused by particles incident at different portions of the scintillator. The signature of an event was a coincidence between PlP2 and DlA, D2A, D3A, or between PlP2 and DlB, D2B, D3B or both within 8 ns. Delay cables were inserted bewteen the counters and the coincidence circuits to compensate for the unequal flight times required for the desired particles to reach each counter. The proper lengths of delay cable were determined by considering the time necessary to reach each counter after passing through the successive intervening media. These values were checked by filling the target with liquid hydrogen and measuring a known point in the proton-proton differential cross section. Standard commercial modules were used for the electronic logic circuitry. A block diagram of the logic circuitry is given in Fig. 24. The details of each block unit are described by Sugarman et al.68 The duration of a beam pulse (called the beam spill) was typically 200 ms long with very little structure. A gating circuit prevented the recording of events whenever an anomalously large flux density occurred via a trigger pulse from the C counter. Pulses from each channel were split to provide scaler records of 52

TABLE III COUNTER DIMENSIONS Counter Width x Height x Thickness P1i 3.5" x 5" x 1/8" 2.5" x 3" x 1/8" 2.0" x 1" x 1/8":P2 ~5" x 7" x 1/2" 4" x 2" x 1/2" D1A 5.5" x 8" x 1/2" D1B -5.5" x 8" x 1/2" D2A 13" x 19" x 3/8" D2B 13" x 19" x 3/8" D3A 17" x 30" x 1/2" D3B 17" x 30 " x 1/2" C 8" diameter x 1/2" Ml 1" x 11" x 1/4" M2 1i' x 1" x 1/4" M3 1" x 11 x 1/4" S 1.5" x 1.5" x 3/8" 53

BLOCK DIAGRAM OF ELECTRONICS Ds1 T L L ~T T T T p ODECE INCIDENCEI COlNCIDENCE I ONCIDENCE TO CALERS I.P I I PD ~CHANCE PD ONCE oCOICIDENCE' GOINCIDENCEI I~N.~.ENCE TO SCALERS V DISCRIMINATOR TO SCALERS SLITTER ~ &7 MIXER Fig. 24. Electronic logic circuitry. Variable delay cables inserted at positions marked by T.

the P channel, the D channels and their combinations. Accidentals were continuously monitored by delaying a portion of the pulse from the D channels by 50 ns with respect to the desired time for a real event. The clipping lines were selected to give a time resolution of 8 ns for an elastic event. This was slightly larger than the time spread expected due to straggling of the particles before reaching the final counters. The chance coincidence were held to less than 10% of the actual events by regulating the beam intensity. Two scalers were used to record the events to provide evidence of any malfunction within the scalers. The measurement of a single point in the differential cross section proceeded in the following manner. The P telescope and the H206 magnet carriage supporting the D1 and D2 counters were positioned using the theodolite. The D3 counters were positioned at the desired place on the parallel I-beams. During this time, the target was being filled with liquid deuterium. The delay cables were selected and connected for this kinematical setting. The target was then bombarded until the desired number of events were recorded and printed out from the scalers. The reading of the M monitor scaler depended upon the contents of the target. The reading of the S scaler had a negligible dependence on the target contents because less than 2% of the beam interacted with the target and the S telescope monitored the beam particles scattered by the C counter in the beam line. The target was emptied, and the target was again bombarded until the S scaler indicated that an equal number of protons had passed through the target. The procedure was then repeated for the next point. Several times during the measurement of the differential cross section at a given energy, polyethylene foils were irradiated for normalization purposes. 55

VI. NORMALIZATION OF CROSS SECTIONS The M and S telescope were employed to normalize the differential cross sections. The half life of C11 20. 5 minutes, makes the reaction C12 (p,pn) Cll convenient to study with this apparatus. A beam of 1-5 x 109 particles per pulse irradiated a 4 mil polyethylene foil mounted on the downstream end of the target. The decay rate of the C11 atom was then measured in a NaI well-counter by detecting the gamma rays produced by pair annihilation of the emitted positron. The rate of producing C11 atom in the foil may be expressed by dN _ dt M where a = cross section for C11 formation, p = flux of protons/sec. through foil, N = Avogadro's number, p = density of foil in g/cm2, M = molecular weight of polyethylene (CH2)n, Nll = number of C11 atoms. These C11 atoms are lost in two principal ways during irradiation of the foil. Some of the molecules containing C11 are scattered out of the foil. This process is termed "hot atom" loss, and has been measured by Cumming et al.69 The fraction of C11 atoms scattered from the foil, a, reduces the rate of producing C11 atoms in the foil to dNll (1 - a) N dt M According to Cumming et al.,75 a 4 mil foil has a value of a =.142. During irradiation some of the ll1 atoms are lost by decaying. The radioactive decay law gives this as dNl dt - KN11(t) where K is the decay constant. Hence the rate of Cil accumulation in the foil is dN11 = \.Np _ -;a-;

with the boundary condition that Nll = 0 at t = 0; the equation is easily solved to yield, Nil = (1 - a) cpNp (1 - e-Kt) (1) The foils were irradiated for 1-minute, 2-minute, and 3-minute periods. Let the irradiation period be called tl. After termination of irradiation, the number of C11 atoms in the foil is Nil = N11 (tl) e -K(t - tl) where t > tl. The number of C11 decays, D, were counted with the NaI well-counter in one minute intervals. Denoting the initial time of an interval by ti and the termination by tf, the number of decays may be expressed by tf Kt D = - KN(t)E dt = Nll(tl) e E(e -e tf) (2) ti where E is the efficiency of the well counter. The total flux, F, is the quantity desired. The Nll(tl) is given in Equation (1) by setting t = t1. Thus substituting into Equation (2) with cp = F/t, one obtains D = E(l - a)aNp F eKt1 - e Ktl)(e Kti -Ktf) KMt1 or -Kt1 KMtlD e E(1 - a)aNp(l - e-Ktl)(e-Kti-e Ktf) The total cross section of the reaction C12(p,pn)CIl, a, has been measured at several energies. In Table IV the values of a are tabulated from 1 to 28 GeV incident protons. The values in our energy range were interpolated from the data of Poskanzer et al. and Cumming et al. Since the half life of C11 is known, the decay constant, K, equals 0.564 x 10-3 sec-1. The monitor counters were calibrated by using F/M and F/S as standard radios for target full and target empty runs respectively, where M is the number of coincidences in the M monitor and S is the number of coincidences in the S monitor during irradiation of the foil by F protons. The statistical error in D and the error in determination of well-counter efficiency made the value of F uncertain by + 8%. Three or more foils were irradiated for each energy, and the normalizations always agreed within 4%. 57

TABLE IV CROSS SECTION FOR C1 (p,pn) Cll Tp Reference p U(mb) 1.0 26.6 + 1.3 70 2.0 26.2 + 0.9 71 3.0 26.8 + 1.0o 71 3.0 29.5 + 1.6 72 4.5 27.4 + 1.4 72 6.o0 29.5 + 1.6 72 9.0 26.2 + 1.5 73 28.o 25.9 + 1.2 74 58

VII. DATA CORRECTIONS A. NUCLEAR ABSORPTION While traversing the target, counters and air, a small percentage of the scattered protons and deuterons were lost due to nuclear interactions. The fraction of particles lost during passage.through a single medium may be calculated from S1 1 ~Np eX dx a = M So = ~1 D, 0 where So = number of incident particles, S1 = number of surviving particles, ~1 = path length in medium N = Avogadro's number, p = density of medium M = molecular weight,'C = total cross section Losses for a series of F media may then be expressed by S- =1' fx. e -~x. l.e -lXl dxl...dxf The integrals may easily be performed to yield s1 fA recorded event is characterized by both the scattered proton and its corresponding deuteron completely traversing their respective telescopes. The probability of proton surviving nuclear interaction is stochastically independent of the probability that a deuteron survives. The probability of two stochastically independent processes occurring equals the product of their individual probabilities. Therefore the probability of both scattered particles surviving nuclear interaction, P, equals 59

/ f _Pk-m /6 p =...1- j-[ 1 - e U e k=l klk J iik The ik and tj in the above formula must be determined. The values for N, p and M may be found in standard tables. A first approximation to the total cross sections, c would be the geometrical cross section. A geometrical consideration only yields a = r (roAl/3)2 where A equals the mass number and ro is the Bohr radiuso In the energy range investigated here, the deBroglie wavelength is sufficiently small to allow one to consider the nuclear scattering as occurring between the incident proton or deuteron and individual nucleons of the nucleus. The cross sections for proton-nucleon and deuteron-nucleon collisions are known over the desired range.76 The energy,-dependence of the cross sections has been taken into account although the percentage of particles surviving changes only as much as 0.5% for the P-telescope and 2% for D-telescope in the energy range considered. The cross sections on materials with A > 1 can then be approximated by aDA = 9DNA2/3 for deuterons and UpNA2/3 for protons. The resulting formulae for ik and Ad are thus NPk a 2/3 for deuterons ~k = -pN Np. 2 and N A3 for protons. The effect of nuclear absorption was tested by placing a slab of lucite.25 in.'thick before the P1 counter. The differential cross section at several points was measured with and without the lucite slab and no statistically significant difference was observed. Using the above formula, the percentage of protons surviving nuclear absorption without the slab was shown to be greater than the survival with the slab by magnitudes varying from.30% to.50% which was well below our counting statistical error. B. COUNTER EFFICIENCY AND DEAD TIME The P1, P2, and D1 counters were essentially 100% efficient. However the larger size of the D2 and D3 counters introduced a slight inefficiency. The two photomultiplier tubes on the D3 counters served to reduce the inefficiencies of these counters significantly. Although the efficiencies of the D2 and D3 counters were not measured directly, they have been estimated to be 99o + 1% efficient. The6counter dead time is the time between the incidence of a particle in 60

the scintillator and the time.that the subsequent electronics has recovered sufficiently to record the incidence of another particleon the scintillator of the counter. The dead time in a given pulse depends upon the length of the beam spill, the number of particles incident per pulse, and the time resolution of the-electronic logic. In this experiment, the beam spill was typically 150 ms, and the singles rate on a given counter was usually less than 8 x 104 counts per pulse. Since the time resolution of the logic circuitry was 8 ns, the counter dead time was less than 1%. If one considers the nonuniform structure of the beam spill, more Jof' the true coincidences may be lost because the losses increase in proportion to the counting rate. The singles counting rates of each counter were not monitored continuously because.they changed with each pulse. Consequently the correction for anomalies in the beam spill are only approximated. The uncertainties in the counter dead time have been included in the systematic error corrections to the data. Co BEAM ATTENUATION The number of protons in the.beam is not constant because many of the protons interact while.transversing the target. Since the downstream end of the target is exposed to only a fraction of the initial beam intensity, N, the total target is exposed to an effective beam intensity Neo The initial and effective beam intensities are related by L N = e Px dx e L where L equals the target length, p equals the density of deuterons in liquid deuterium, and a equals the total proton-deuteron cross section. The point x = 0 corresponds to the upstream end of the target. The effective beam intensity was 0.6* less than the initial beam intensity. Do ANGULAR UNCERTAINTY The finite width of the P1 counter, w, and the length of the target, L, introduce an error in the polar angle.. The particles from the target entering the P1 counter may have angles as large as 9 + AGp or as small as 9 - An where -Ap = tan-l Lw + L sin e] L2r - L cos A = tan l rw + L sin e AGn 2r + L cos 6 61

with r equal to the distance between the counter and the target. Transforming the AGp and Aen into the center-of-mass frame, one obtains the expression!?~~.~l~f! 1),tan2 t + 1 using tip n=. p usinpngn" -B (p,nThe values of cos 8*, e*, A*P, and Ae* are given in Table V for the various angles and counter distances used in this experiment. Eo MULTIPLE COULOMB SCATTERING -The sizes of the counters were designed to eliminate.loss of events due to multiple Coulomb scattering. This requirement was checked during the experimental run by substituting a smaller P1 counter for the original P1 defining counter at given points. No significant difference was measured between using the designed subtended solid angle and the smaller solid angle, which would confine the corresponding particles in the D-channel to a smaller portion of the D counters and thus minimize the losses. This technique could not be applied to determine the Coulomb scattering losses sustained by the P1 counter itself. To a first approximation the number of particles scattered out of the.line of coincidence with the P telescope is equal to those scattered into the telescope. Due to the energy of the particles and the distances involved, less than 1% of the coincidences are estimated to have been lost through multiple Coulomb scattering. Fo ACCIDENTAL AND BACKGROUND COINCIDENCES Independent particles in the deuteron and proton channels may produce pulses which would be recorded as an event if they occurred within a short time interval. These chance coincidences were continuously monitored by recording the coincidences between the P and D channels when they were 50 ns out of time. The length of this delay cable was varied over a range of 30ns with no appreciable difference in the percentage of accidentals for a given run. The percentage of chance events was always kept below 10% by adjusting the beam intensity. In determining the number of good events, the number of chance events was subtracted whibch;resultie.d nia larger statistial.e.rror to compensate for the presence of undesireable coincidences. For each point of the differential cross sections, the measurement was made for a full target and an empty target. The S monitor counters provided 62

TABLE V UNCERTAINTY IN SCATTERING ANGLES Cos * 6* Pe 0n T =1.0 GeV P -.885 152.25 3.03 3.07 -.875 151.04 3.07 3.11 -.850 148.21 3.33 3.37 -.825 145.59 3.61 3.65 -.800 143.13 4.20 4.24 -.750 138.59 4.31 4.34 -.700 134.43 3.84 3.85 -.650 130.54 3.66 3.66 -.600 126.87 3.69 3.69 -.550 123.37 3.34 3.32 -.500 120.00 3.15 3.13 T =2.0 GeV.565 55.60 1.41 1.39.600 53.13 1.40 1.38.650 49.46 1.37 1.36.700 45.57 1.35 1.33.750 41.41 1.32 1.31.800 36.87 1.29 1.28.825 24.41 1.28 1.26.850 31.79 1.26 1.24.875 28.95 1.24 1.22 63

TABLE V. CONT'D Cos e* e0*' 60 A *0~ * p n Tp=1.3 GeV -.895 153.51 3.16 3.21 -.875 151.04 3.45 3.50 -.850 148.21 3.79 3.84 -.825 145.59 4.15 4.19 -.800 143.13 4.54 4.59 -.750 138.59 4.34 4.36 -.700 134.43 4.13 4.14..650 130.54 3.92 3.92 -.600 126.87 3.73 3.72 -.550 123.37 3.37 3.35 -.500 120.00 3.38 3.36 -.460 117.39 3.18 3.16 Tp =1.5 GeV -.900 154.16 3.14 3.19 -.875 151.04 3.46 3.51 -.850 148.21 3.56 3.60 -.825 145.59 3.89 3.93 -.800 143.13 4.24 4.28 -.750 138.59 4.36 4.38 -.700 134.43 4.16 4.16 -.650 130.54 3.95 3.95 -.600 126.87 3.75 3.74 -.550 123.37 3.39 3.38 -.500 120.00 3.20 3.18 -.450 116.74 3.21 3.18 64

the means to determine when the amount of incident beam for the empty target equalled that for the full target. The average percentage of recorded target empty events to target full events for all runs was 1.3%. Of these target empty events, a o.1 contribution was made by the residual deuterium vapor present in the target when the liquid deuterium was removed. The full target does not contain the vapor, consequently this contribution was not subtracted from the true coincidences. The resultant 1.2% background were caused primarily by interaction with the carbon atoms of the mylar surrounding the target. The number of events for each point of thecdifferential cross section was corrected for background events by subtracting 1.2% as quasi-coincidences to yield a net corrected number of PD coincidences.

VIIIo RESULTS OF EXPERIMENT The differential cross section at a given angle was calculated by the formula do Y abcdgh dQ pLAJ (AsQ) where Y = (E- Ea) full tgt. - (E - Ea) empty E = number of events Ea = number of accidental events p = density of liquid deuterium in deuterons/cc. L = length of deuterium target F - normalization factor for M monitor M = number of coincidences in M monitor J = Jacobian transforming solid angle from laboratory to c.m. system \A2 = solid angle subtended by P1 counter a = correction for nuclear absorption b = correction for counter efficiency c = correction for counter dead time d = correction for multiple Coulomb scattering g = correction for beam attenuation h = correction for background events The results are shown in Tables VI and VII and Figs. 25-26, The error bars are those due to counting statistics and range from 3% to 10. The total error from nonstatistical sources is 10o of which 81 is due.to normalization error and 2% is due to the error in the Jacobian introduced by the uncertainty in beam energy. 66

TABLE VI BACKWARD PROTON-DEUTERON ELASTIC SCATTERING dco/dQ c.m. Lb/ster Cos e* 1.0 GeV 1.3 GeV 1.5 GeV -.goo 2.05 +.21 -.895 4.50 +.33 -.885 12.99 +.42 -.875 10.93 +.36 4.07 +.35 1.73 +.17 -.850 8.80 +.29 3.47 +.26 1.06 +.12 -.825 7.99 +.26 3.39 +.23 -.800 6.82 +.22 2.22 +.11 0.76 +.08 -.750 4.52 +.15 1.49 +.15 0.63 +.o06 -.700 3.30 +.11 1.08 +.11 0.49 +.04 -.650 2.84 +.09 1.00 +.10 0.30 +.03 -.600 2.70 +.12 0.80 +.08 0.34 +.03 -.550 2.15 +.13 0.56 t.06 0.29 +.03 -.500 1.99 +.09 0.66 +.06 0.29 +.03 -.460 0.50 +.05 TABLE VII FORWARD PROTON-DEUTERON ELASTIC SCATTERING do Cos e* -td c.m. at 2.0 GeV.875.4422 78. o9 + 2.52.850.5306 64.65 + 1.94.800.7075 51.55 + 1.55.750.8843 36.42 + 1.09.700 1.0612 22.79 + 0.68.650 1.2381 12.18 + 0.37.600 1.4149 7.11 + 0.21. 565 1.5387 4.97 + 0.15 67

20 I$.Thi Ex winnt d b $tr. LO rEV. I. $ GEV. 5 -: - - 7 -.8 -.9 - LO COS e' Fig. 25. Experimental differential cross sections at 1.0, 1.3, and 1.5 GeV in the center-of-mass system.

mlopoootI 100,000 _, 2.0 GEV -" =oThis Experiment Kirillova efo.: Optical Point 1,000 do dS, cab Str. 100 50 I0 5 K 2.4.6.8 LO 12 14 1.6 18 2.0 -t (GEV/ C)2 Fig. 26. Experimental differential cross sections a.t 2.0 GeV in the centerof-mass system. 69

The curve in Fig. 25 appear approximately exponential in character. In low four momentum work many found it convenient to express differential cross sections as exponentials in four momentum transfer. In Fig. 27 the backward differential cross sections have been plotted as an exponential to a polynomial in T, where T equals the four-momentum transfer minus the four-momentum transfer at 1800. A chi-square test of the powers of tau up to the fourth power specified a quadratic as the best fit. The differential cross sections were then written as d = e(a + bT + cT2) fib dT (GeV/c)2 where the values of a, b, and c were determined by a least square procedure and are displayed in Table VIII. TABLE VIII COEFFICIENTS IN EXPONENTIAL FIT Tp a b c (GeV) (GeV/c )2 (GeV/c )4 1.0 -4.97 6.48 -3.54 1.3 -3.58 4.47 -1.49 1.5 -2.66 4.68 -1.82 70

900 5 IDGEV so 1.5 GEV 5I. do (b 9.00. o.12.4.6. 1.0 i.2 1.4 r a(GEV/C)2 Fig. 27. Exponential fit to cross sections at 1.0, 1.3, and 1.5 GeV. 71

IX. CONCLUSIONS The data of'this experiment agreed well with the point of the differential cross section obtained by Bayukov et al.) at 1.0 GeV. At present, there are no other published data in the few GeV range with which to compare the backward proton-deuteron elastic differential cross sections obtained in this experiment. The 2.0 GeV small angle data of Kirillova et al.15 described a strict exponential trend for four-momentum transfers below 0.13 (GeV/c)2. The experiment observed a shoulderlike departure of the forward differential cross section from the original slope of the diffraction peak as the four-momentum increased above 0.44 (GeV/c)2. The backward scattering of protons by deuterons has received reasonable interpretation by assuming that one-neutron exchange is the dominant process. The slope of the observed backward peak was predicted by the one-nucleon Feynman diagram using the proton-neutron-deuteron vertex function of Blankenbecler, Goldberger, and Halpern,l6 but the predicted magnitude exceeded that of the observed peak by a factor of 5.00 at 1.0 GeV incident proton kinetic energy. The use of various form factors reduced the magnitude considerably and also adversely affected&the shape of the cross section. The form factors were the Fourier transforms of the deuteron wave function in coordinate space, and their original purpose was to obtain an accurate representation of the high momentum components of the deuteron. The form factors suggested by the regular Hulthen wave function, the Hulthen wave function with a hard core and the Moravcsik analytic fit to the Gartenhaus wave function were presented in this work, and each of the form factors was shown to decrease the agreement of the theoretical differential cross section with the experimental data. Apparently, the high momentum components of the deuteron are somewhat larger than previously suspected. A consideration of absorption in the initial and final state interactions may lead to a modest reduction of the theoretical magnitude and to a possible change in the shape. However, due to the lack of proton-deuteron phase shift data in this energy range, the usual methods of computing the absorption cannot be applied, Calculation of the phase shifts from the scant forward data was considered, but the application of the resulting absorption factors to a partial wave expansion of the backward scattering amplitudes could not be justified. Ross and Shaw77 have pointed out that the suppression of the differential cross sections due to absorption may differ greatly in the forward and backward directions. The backward differential cross sections were also analyzed within the framework of three-body formalisms. The problem was formulated in the method of Amado,39 and the zeroth iteration (the one-nucleon exchange diagram) was found to exhibit the closest agreement with the available data. Since this method utilized the low energy parameters of the effective range theory, a nonrelativistic propagator was used, The difference in the predicted dif72

ferential cross sections between using a relativistic propagator and a lonrelativistic propagator is shown in Fig. 28. The relativistic propagator predicts a slightly steeper slope, which more closely fits the experimental data. The predictions of the three-body method may be improved by solving a completely relativistic set of Faddeev equations. The shoulderlike departure of the data in this experiment at 2.0 GeV from the forward diffraction peak has been interpreted by extending the high energy approximation of Franco and Glauber66 to higher momentum transfers. Consideration of single-scattering only does not given a satisfactory interpretation of the data above four-momentum transfers of 0.5 (GeV/c) 2; however, inclusion of double-scattering interactions is observed to account for the change in slope of the differential cross section. The magnitude and slope of the experimental data for four-momentum transfers above 0.5 (GeV/c)2 are determined quite accurately as manifestitations of the incident proton being successively scattered by both constituents of the deuteron. Due to interference between single- and double-scattering, a dip occurred in the predicted differential cross section which has no apparent justification in the experimental data. The depth of the dip has been shown to be sensitive to the sign and magnitude of the real parts of the scattering amplitudes. The large uncertainty of the ratio of the real part of the proton-neutron scattering amplitude to the imaginary part of the scattering amplitude, an = 0.2 + 0.4, determined by Kirillova et al.15 and the predictions of the high energy approximation allow one to assert that an is negative as is the case for proton-proton elastic scattering at 2.0 GeV. 73

35 30 1.0 GEV 25 2 -10 do i /:COS - 1 5 @~~Str. /e15 I0 ~~~~74~ ~0 -.4 -.5 -.6 -7 -.8 9 -1.0 cos 8* Fig. 28. The effects of relativistic and nonrelativistic propagators at 1.0 GeV. (I) Using relativistic propagator; (II) using nonrelativistic 74

APPENDIX A EVALUATION OF MATRIX ELEMENT The square of the matrix element with sums over the proton and deuteron spins is evaluated. In Chapter II, the matrix element was expressed by M = u(p)T 2.17 + m + i i (p2) Then (n2 + M2)2 ( + [ u(pl)T r., ++ m) rl u(p2)T The expressions for rl and F2 are given in Chapter II. Using the identity 2I + m) _4_ j~, - it o l T(p2).(p _)'t - t r -;- =3 0t- IU(P2) 74unpl) |Y= tr 74 2m 74 2m spins and summing over the proton spins yields Z I MVM -Z 1 1 1 eutns protns deut. (n2 + m2)2 (2im) 2 (2-UMd)4 spins'pins spins tr [(2 % ~1;) (Md i9l) (-iY + m) (Md - iI2)(-.t2)(il/ + m) (2 ~2)(Md - iA2))(-i~ + m)(Md - i2)(Y'l)(i i'2 + m)] The deuteron spins are summed over using the identities Z (P o )(n o ) = p o + pd)(nd) deut. o 2 spins d 75

and it + i = 2 p n for any four vectors p and n. The matrix element then assumes the form 1L IM2 3 (2mMd - pilod2)2 (2 2 6 128 md (n + m2)2 (m)4 SF(] ) K where K = [m2Md - m2d o d2 - ndl d2 - mMdn (d1 + d2) + 2n dl1 n o.d2 - d n o n 76

APPENDIX B DEUTERON WAVE FUNCTIONS This appendix lists the various deuteron wave functions used in present theoretical work. A basic requirement of the wave functions is that they adequately determine the static properties of the deuteron. If a is the same as defined in Chapter II with u(r) as the SUstate wave function and w(r) as the D-state wave function, the non-relativistic deuteron problem with a tensor force consists in solving the coupled differential equations: ud2 [K Vc(r)] u(r) + 22 vT(r) w(r) = 0 d wr)(-)+,(v w(r).2Wv(r)u(r) =-0 2 dr2 r c (r) + 2vT(r)] wr) + 2 v (r) O safisfying the conditions that u2r) + w2(r)] dr = 1 and the wave functions go to zero as r goes Eo zero or infinity. The vc(r) is the central potential and the vT(r) is the tensor potential comprising a total potential, V(r) = -m vc(r) + S12vT(r)] where ~.:= 1. Once the wave functions u(r) and w(r) are determined, the electric quadrupole moment may be calculated by A 1 "o r2 2 Since the electric quadrupole moment is known to be positive, the u(r) and w(r) must have the same sign. If the proton magnetic moment ~p and the neutron magnetic moment En are known, the deuteron magnetic moment is given by 00 d = p + n 23 (p + Ln -w) (r)dr The following deuteron wave functions are frequently used in practice. 77

lo Hulthen wave function28: (a) Regular wave function u(r) = N(e-ar - e Pr) a = 0.232 f-1 fi = 5,18 (b) Wave function with radius rc hard core [1r e7(r-rc) u r) N e' 1 e wry~) = N-er [1 e 7(rrc) r 1+ 3(1, e 7) + 3(- e-r)@ Lr a2r2 where all positive n are allowed for rce 0 and ~ and y are determined'to fit QJd and other properties of the deutrono 2e Gartenhaus wave function30: A numerical tabulation for the deuteron wave function was computed by Gartenhaus using a cutoff Yukawa theory~ Moravcsik obtained analytic fits to the tabulation in the form: u(r) = N(e-r -e -dr)(1 - e-cr)(1 e-g) 0.o658 r3 for 0 < r < 0 63f w(r) = 2.34r3e-2r for 0o63 < r < 2l10f 0o147e-0256r + 00810e-O~577r-r for 2010 < r <+ oo where c = 6o853(, d = 80190a, g = 10o776a 3o Hamada-Johnston wave.function78 The deuteron wave function with a hard core of 0-343f was tabulated by Hamada and Johnston, who assumed a potential with central, tensor, linear LS and quadratic LS termso The long range.quadratic LS potential was added to decrease the magnitude of the 3D2 phase shift at high energieso An analytic fit to the Hamada-Johnston wave function without the hard core has been constructed by McGeeo79 The Sand D-state wave functions are of the form 78

4 u(r) = N(e-ar + Z C e-ejr) j=l j w(r) = pN arh2(iar) + jl Djrjrh2(iir) where h2 (iy) is the spherical Hankel function and the values of Cj, Ej, Dj, and ~j are given in the following table: u(r) w(r) Cj ~j Dj.j 1 -. 6361 5.732a -20.34 4.833x 2 - 6.6150 12.844a -36.60 10.447a 3 15.216 17.331a -123.02 14.506a 4 - 8.9651 19.643x 305.11 16.868x 5 -126.16 21.154m 4. Gaussian wave function: A convenient analytical form of the deuteron is the simple Gaussian. By using a variational method to minimize the energy for the deuteron ground state, Verde8 obtained the form 2(r) = 2)P3/4 _re2r2 U(r) = ( — — re where f = 1.3367a 5. Christian-Gammel wave function: The deuteron wave function has been fit to a sum of Gaussians by Christian and Gammel.81 They obtained a wave function of the form u(r) = 0.02133r e-0'03r2 + 0.08582r e-0l16r2 + 0.18115r e-0.76r2 79

The above..are the fnbst:: ooimonly utedl.analytid.al'fo-tm$ ofs the dbuteron wave functiono There also exists many deuteron wave functions in tabular form only which many researchers use in numerical calculations. 80

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ACKNOWLEDGME NTS I am grateful to Professor Oliver E. Overseth for his inspiring guidance during the course of this work. As a research physicist, teacher, and friend, he has made my doctoral studies a very satisfying experience. I would like to thank my colleagues Dr. Richard M. Heinz and David E. Pellett for their indispensable assistance during the experiment. Professor Lawrence M. Jones deserves my deepest appreciation for many helpful discussions; and I wish to thank Dr. YingYeung Yam for interesting conversations on the theoretical aspects of proton-deuteron scattering. 86

UTnclassified Security Classification DOCUMENT CONTROL DATA - R&D (Security claseificatton of title, body of abstract and Indexing annotation must be entered when the overall report is clasailied) 1. ORIGINATIN G ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan Unclassified Department of Physics 2 b GROUP Ann Arbor, Michigan 3. REPORT TITLE PROTON-DEUTERON ELASTIC SCATTERING AT HIGH MOMENTUM TRANSFERS 4. DESCRIPTIVE NOTES (Type of report and Incfusive datee) Technical Report No. 25 5. AUTHOR(S) (Lest name. first name, initial) Coleman, Ernest 6. REPO RT DATE... TOTAL NO. OF PAGES 7b. NO. OF REPS July 1966 85 81 8a. CONTRACT OR GRANT NO. 9.. ORIGINATOR'S REPORT NUMBER(S) Nonr-1224(23) 03106-25-T b. PROJECT NO. NR-022-274 c. |S b.OTHER REI PORT NO(S) (A ny other numbers that may be assigned this rebport) d. Technical Report No. 25 1 o. ^ v A IL ABLITY/LMITA*tION NOTICES Distribution of this document is unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Department of the Navy Office of Naval Research Washington, D. C. 13. ABSTRACT An experimental and theoretical investigation of proton-deuteron elastic scattering at high momentum transfers is presented. The differential cross sections for backward elastic scattering at incident proton kinetic energies of 1.0, 1.3, and 1.5 GeV have been measured for four-momentum transfer squared (-t) from 2.6 to 5.0 (GeV/c)2, which corresponds to cosine of center-of-mass proton scattering angles (cos Q*) from -0.5 to -0.9. A backward peak is observed, and the slope and magnitude of the peak have been determined. At 2.0 GeV for forward elastic differential cross section has been measured for -t from 0.44 to 1.54 (GeV/c)2 or cos G* from 0.875 to 0.565. A shoulderlike departure from the forward diffraction peak was observed. The one-nucleon exchange peripheral model has been successful in interpreting the backward peak. Calculations based on modern three-body quantum mechanical formalisms for the three-nucleon system supporting a two-body bound state also suggest the one-nucleon exchange process as the dominant mechanism. The measured forward differential cross section has been explained by the importance of doublescattering of the incident proton at higher momentum transfers. A negative value for the ratio of the real part to the imaginary part of the neutron-proton elastic scattering amplitude at 2.0 GeV is shown to yield maximum agreement with the exerimental data. D 1JAN64 1473 Unclassified Security Classification

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UNIVERSITY OF MICHIGAN 31 90111111111111111 115 029I 511118611 3 9015 02947 5186