Analysis of intermediate energy nucleon-deuteron elastic scattering

ANNALS

OF

PHYSICS

82, 189-247 (1974)

Analysis of Intermediate Energy Nucleon-Deuteron Elastic Scattering E. A. REMLER AND R. A. MILLER* Department

of Physics,

College

of William

and

Mary,

Williamsburg,

Virginia

23185

Received January 25, 1973

We present a theoretical analysis of a broad range of aspects of intermediate energy nucleon-deuteron scattering. This analysis is based on a multiple scattering approach using knowledge of the deuteron’s structure and nucleon-nucleon interactions. Conversely, comparison of this theory with experiment can yield information about low and intermediate energy strong interactions. The relationship of this multiple scattering type of approach to the complementary Faddeev equation approach is discussed. Our program consists of calculating the single scattering and one nucleon exchange contributions in a realistic way then parametrizing the remaining contributions as an S-wave. We argue that the largest error in this analysis is the P-wave part of the double scattering and we give estimates of its size. The single scattering integral is evaluated numerically. Coulomb effects are neglected. We derive the relativistic expressions for single scattering and nucleon exchange and discuss the approximations made, including the off-mass-shell extrapolation of the nucleon-nucleon scattering amplitude. Fits are made to experimental measurements of differential cross sections, nucleon polarizations, and total elastic cross sections. Unitarity is maintained. We tabulate the partial waves for J < 5/2, L < 2. They are consistent with recent Faddeev calculations. We argue that with the additional calculation of double scattering the deuteron D-state percentage can be determined to the same relative uncertainty as the differential cross section. Even without the calculation of double scattering, our results indicate a D-state percentage around 8 %. In an effort to provide benchmarks for future work, we have tried to be conscientious in describing our techniques and in tabulating numerical results. Comparisons are also made with earlier analyses.

I. INTRODUCTION

This paper reports a theoretical analysis of intermediate energy nucleon-deuteron elastic scattering experiments. Theoretical input into such an analysis is, in broadest terms, knowledge of the strong interactions. In the present context this interaction must be known in two distinguishable energy regions. First, the deuteron’s internal *Present 08903.

address: Physics Department,

Rutgers University, 189

Copyright All rights

0 1974 by Academic Press, Inc. of reproduction in any form reserved.

New Brunswick,

New Jersey

190

REMLER

AND

MILLER

hadronic structure must be known. This corresponds to baryon number equal two vertices with direct channel energy at the deuteron pole. Second, one must know about the interaction of, in general, two hadrons at a range of higher direct channel energies corresponding to scattering of the incident nucleon by all the various possible constituents of the deuteron. Conversely then, analysis of such experiments should yield information about the basic strong interactions in these two sectors. Two practical theoretical approaches to this problem are being used at present. These can be viewed as being complementary approximations to the same approach. Both consider the elastic amplitude as part of the solution to a many body linear integral equation. The kernel of this equation and the inhomogeneous term (the deuteron wavefunction) constitute the basic strong interaction physics under investigation. People utilizing Faddeev type equations [ 1,2] approximate the kernel to a degree sufficient to be able to solve the resulting integral equation. This usually involves, given the present state of the art, separable approximations to the first few partial waves of the nucleon-nucleon interaction and working in a purely nonrelativistic context. People utilizing the alternative multiple scattering series approach in effect deal with the kernel as carefully as they know how but solve the integral equation only perturbatively [3,4]. Although this has in the past only been done nonrelativistically it can with equal ease, as we show here, be done entirely relativistically. This consideration is necessary and important towards the higher end of the energy spectrum here under investigation. These mathematical approximations each enjoy their own corresponding physical domains of maximum efficiency. Thus at incident nuclear lab energies of less than say 40 MeV it is clear that the incident nucleon may bounce around many times during the scattering process and interact with each constituent of the deuteron via only a few partial waves. Conversely, it will bounce around only a few times [5, 61 at high energies and interact via many partial waves. Furthermore, it is clear that unless one is at the extreme limits of these two physical situations some combination of these pictures is appropriate along with some corresponding combination of mathematical approximation procedures. From the Faddeev equation viewpoint this leads to the well known idea of breaking up the kernel into two parts, solving the integral equation exactly for one part and then calculating the perturbation due to the other. In recent years steps towards implementing this idea have been taken [7, 81. The work presented in this paper may be conceived as motion towards the same goal from the alternative starting point of the multiple scattering theory. We outline next a more specific theoretical program suited to the analysis of elastic intermediate energy scattering data. Following this will come a description of the initial steps in this direction which are then reported in detail in subsequent sections of this paper.

NUCLEON-DEUTERON

ELASTIC

191

SCATTERING

The incident nucleon laboratory kinetic energy TL , considered here has the range 30 2 TL ,( 200, in MeV. The total transition matrix is decomposed as T = TX -I- Tl i- Ts + A

(1)

where, as shown in Fig. 1, TX is the contribution of the nucleon exchange Feynman diagram, Tl,2 the single and double scattering contributions, and A is all the rest.

X

A+...+=?mp+ +... FIG. 1. Classification

of reactions

in nucleon-deuteron

scattering.

We could include in the definition of TX the sum of all topologically equivalent diagrams of the exchange class, i.e., exchange of nucleon resonances etc., for Tl and T, . However in this energy region, given current theoretical and experimental resolution, such considerations are premature. We will not discuss these more exotic mechanisms any further other than to point out that a proper understanding of the ordinary contributions is a necessary prerequisite to the meaningful investigation of the resonance contributions and that this requirement was a major motivation for the present work. Some contributions to A are also shown in Fig. 1. These correspond to high (2 3) order multiple scatterings and three body force effects. The central theoretical approximation we suggest is that: (i) A is most important towards the lower end of our energy spectrum; (ii) in this region it has low partial wave content; (in) it is not at present worthwhile to attempt to calculate 595/g2/1-13

192

REMLER AND MILLER

d at our energies but rather simple to parametrize it using S-waves only (two complex parameters) and find their values by fitting to the data. The contention that d is relatively isotropic when it is of importance is indirectly supported by a number of investigations. Sloan [9] has shown in a model nucleon deuteron elastic scattering calculation that all partial waves, except the S waves, are accurately taken into account by the multiple scattering series when terminated beyond double scattering. That is to say, that the multiple scattering contributions to d are isotropic at about a 1 % accuracy level. We have verified [lo] that the Born terms used in Sloan’s model are of reasonable order of magnitude so that his estimate is valid. With respect to three body force contributions to d one can characteristically say little. Estimates usually turn out to be small and since they tend to be short range they should in any case be adequately covered by an S-wave parametrization. At very high energies one can point to Glauber type considerations and calculations which are equivalent effectively to neglecting d entirely [5, 61. Finally, one of us had previously reported some success in simultaneously accounting for both 600 MeV and 146 MeV elastic p - D data in the forward hemisphere using an approximation to single and double scattering only [I 11. This model is quite restrictive and therefore predictive in nature. Although at each energy one has two complex parameters to fit data with, these are all attached to isotropic additions to the transition matrix and describe restricted spin dependences. These can consequently affect theoretical curves in only a limited fashion while at the same time there are many nulceon-deuteron observables besides cross section and polarization to be simultaneously fit. Consequently most of the task of agreeing with experiment lies directly on those terms computed directly from basic strong interaction theory. The analysis presented in this paper falls one very large step short of the one first described. Specifically, due solely to human and computer time and energy limitations, T, is not computed. The procedure makes the approximation TrTxi-T,i-Dgi-Q~=Tx,,,,

(2)

where TX and Tl are as before, 9 = (1/3)x’* * 7c + (l/3) ix’* 9 = (2/3) x’* - x - (l/3) ix’*

x x * 0, x x - a,

(3) (4)

these being doublet and quartet projection operators in the combined spin space of the deuteron and nucleon, and D and Q are constant (S-wave) complex amplitudes fitted to the data.1 Thus the S-wave part of T2 is combined with that of A. The main error is then in the unaccounted for P-wave component of T, (all partial 1 Our notation will be explained in detail in the next section.

NUCLEON-DEUTERON

ELASTIC

SCATTERING

193

waves refer to the overall center-of-mass frame). In the forward direction, T is dominated by Tl , the constructive sum of very many partial waves so that an error in only one of them will be relatively unimportant. Based on Sloan’s estimates we expected errors in cross-sections of the order of 40 % especially near 180” and this is born out by our calculations. In relation to previous work done in this area based on the multiple scattering approach [3,4] we have made three technical advances. First we calculate the single scattering numerically. Second we add in S-wave phenomenologically, our partial realization of the tremendous importance of multiple scattering at these energies. Third, we calculate relativistically so that the kinematics at higher energies is correctly accounted for (relativistic dynamic effects are quite another matter). These technical advances reveal a clear and comprehensive theoretical understanding of the data over a wide energy range. It is wide especially in that it makes the transition from essentially nonrelativistic to essentially relativistic theory and experiment. In Section II we derive the formulas used for the evaluation of TX and Tl and discuss the theoretical approximations made. This can be safely skipped by readers who are not theoretically inclined. In Section III we describe our fitting procedure and graph data, fits and various theoretical contributions to cross section and polarization. One can then see why and where the theory either succeeds or fails. The most glaring failure in the approximate theory used here is in reproducing the low energy polarization. This is ascribed mainly to the missing P-wave contribution of double scattering. Support for this contention is given in Appendix E where we show results of adding a phenomenological P-wave. At each energy we have performed the angular momentum decomposition of S,,, and S,,,,, where S,,, is the theoretically computed contribution to the scattering matrix due to exchange and single scattering and S,,,,, is that obtained by fitting (see Eq. 2). These results are shown in Section IV. They are useful for a variety of reasons. The “Born” terms (X + l), which we believe are realistic, can be compared directly to the driving terms used in Faddeev equation treatments (TX as a function of angle is also given in Appendix C) and may be helpful for phase shift analyses. We also follow the energy dependence of the S-wave phase shifts via Argand diagrams. This allows all the experiments to be connected and compared as a group. This allows us to identify a probable error in the 40 MeV cross-section data and also a possible disagreement between the 146 MeV and 155 MeV data. It is of course the ultimate purpose of investigations such as this to learn something new about the basic physics. Scattering by deuterons should be particularly useful in this regard in the foreseeable future. We demonstrate these remarks by attempting to measure the deuteron D-state percentage utilizing analysis of the data near 146 MeV. Since our theory is incomplete (no double scattering) our

194

REMLER AND MILLER

measurement is correspondingly tentative but does, we believe, prove the feasibility of such a measurement and indicates the end result. All this is in Section V. A summary of results is given in Section VI. Detailed descriptions of computational procedures are collected in appendices as well as some tables of values of quantities used during intermediate levels of the calculation. In Appendix A the deuteron wavefunction used throughout is described. In Appendix B our calculation of the on- and off-shell nucleon-nucleon amplitudes is described. In Appendix C the single scattering integration is described and results compared with previous calculations using the sticking factor approximation described in Appendix D. In Appendix E an improved fit to the low energy data using nucleon-spin coupled P-waves is shown and discussed.

II. THEORETICAL

FORMULAS

AND THEIR APPROXIMATIONS

We begin with a glossary of basic symbols and conventions used throughout this paper. E and F are used as subscripts designating entering (i.e., initial) and final states respectively. p and D are momentum four vectors for the external nucleon and deuteron. v, p are the magnetic quantum numbers for nucleon and deuteron respectively along z axis, taken perpendicular to the reaction plane and measured in the centerof-mass system. Polarization three and four vectors describing the deuteron are used as follows.

4&l)

= W/%Q(l,

fi, 0)

x(0) = (0, 0, 1).

(5)

~~,~(p) are covariant polarization four vectors whose spatial parts equal Z(P) only when evaluated in the rest frame of DE,F , respectively. Thus in general one has T&)

Bras and kets are invariantly (P’V’

. DE = T+(P)

. DF = 0.

(6)

normalized

I PV> = (PO/M)


- P) &” ,

w

W’

- D) 4/u,

(7) (8)

where M and MO are the mass of the nucleon and deuteron, respectively. These normalizations and the following definitions of the S-matrix, transition

NUCLEON-DEUTERON

matrix and scattering amplitude nonrelativistic treatments [ 141.
IS -

ELASTIC

go over smoothly into those commonly

1 I PA) = -2n&p, + D, - PE T(F; E) = (~T~)-~(W/MM~)~(F; E), uF,E w2

=

195

SCATTERING

=

(PE

+

if@?

-

PE) T(C E),

‘%‘Y

DE)’

=

used in (9 (10) (11)

(PF

+

(12)

DF)~.

As discussed in the introduction, in this analysis we disregard such aspects of the deuteron’s structure as its possible isobaric content and therefore describe it solely via its two nucleon Bethe-Salpeter amplitude written as (0

I T(qLh)

#Cx2>>

I DP>

=

(~1x2

=

4x2~1

I Q-4

(13)

I W

where T is the time ordering operator and # are vectors in spin-isospin Similarly one defines 0% I TO,&3 $(x2>> I 0) = =

-
I ~2x3,

space.

(14)

and we repeat the usual caveat that because of the time ordering operator (15) Using similar notation we write the nucleon four-point function in position space as (0

I T(rGh)

#

$(xs>

16h>>

I 0)

=

Cv2

I 7 I x,x,);

(16)

the two point function as (0

I W&4

$(x2>>

and the single nucleons “Bethe-Salpeter”

(0 I w

I 0) = (XI I -i-I ~2);

(17)

wavefunction as

IPV) = (x I PV>.

08)

Leaving implicit the renormalization constants and neglecting continuum contributions to the spectral representation of the nucleon two point function one has (Xl 1 7 1XJ = zqx, - x2) = i(2n)-4 s dk (y * k - m + iO)-’ e-*k’bl-ze), ( x 1pv> = (274-f+ u(p, v) e-i***.

(19 (20)

196

REMLER

AND

MILLER

Although the general procedure necessary to express the S-matrix for processes involving composite particles in terms of their Bethe-Salpeter amplitudes has been in the literature for many years [15, 161, knowledge of it is not common nor are examples plentiful. We therefore derive next the particular formulas for exchange and single scattering. Let $P(*)(cY/?*.a) be the operator projecting out the single particle contribution to a matrix element of Heisenberg operators in the limit as t, + ts + *a* + &co while keeping relative times tied. Thus in particular we have: ~(+)(1)(x,

I 7 I x3x4) = cwa I QJWP

9’+‘w(~1~2

where we have introduced indices, e.g.,

I 7 I %I = (Xl I PVXPV I x&

the convention

of implied

(21)

I %a9

(22)

summation

over repeated

I PVXPV I = 2 1 (WPO) dP I PVXPV

Y

I-

(23)

Utilizing the weak convergence theorem of LSZ one gets, upon applying this as a preliminary example to the four-point function LP+yq

zY(+‘(l’) Y(-)(2) Lz+)(l)(X,‘X,’

= (xi

IPl’vl’Xx;

17 I x,x,)

I Pel%XPI‘~1IP~‘VZ’

I S I PzV,P,YXP,V,

I x,)
I x1).

(24) Applying Eq. (21) to the left-hand side of Eq. (24) and comparing like terms yields
I

I s I PZV,Pl%)

dxl’ dx,’ dx, dx,
I xl’)
v2’ I Xa’)h’%’ I 7’ I Wlx% I Pz%Xx1 I P& (25)

where 7’ is the sum over all Feynman diagrams with external legs chopped off. All these formulas can be transcribed to momentum space by using the transformation bracket (p I x) = (24-a

eip.2

(26)

which implies, using Eq. (20), (P’ I PV> = (27v

XP’ -P)

U(PV),

(27)

so that one has, quite simply, --2Mpl’

+ pzl -PI

- PZ) T(PI’P,‘; P~PI)

=
I 7’ I PaPA 4P,%) U(PlVl).

(28)

NUCLEON-DEUTERON

ELASTIC

197

SCATTERING

Getting on now to the case at hand we first note s+y3’)

S-)(3)

L.P+)(1’2’) 3+~(12)(X~‘x;x,’

= (XI’XZ’ I D’$>h’

I P’v’W’P’P’V’

17 1x,x,x,) I S I PVQ-+(PV

I XW’P

I WI>. (29)

One now looks at contributions to the six-point function which give nonvanishing contributions in the above limiting procedure. For reasons already discussed in the introduction we limit ourselves to those contributions which can be written in terms of the four point functions only, i.e., contributions which can be classed as being due to relativistic two nucleon forces. Such contributions can be ordered in terms of their number of four point functions. The lowest order nonzero contributor contains one four point function and is given by

(xii’ I -I-I X&l’X2’ I 7 I %X3

(30)

which upon application of the limiting operators of Eq. (29) and using Eqs. (21) and (22) yields the zeroth order (no-interaction) term in elastic nucleon-deuteron scattering. Exchange scattering arises from one term in the next order contribution, s

dx,” dx,” [ -(x1’x2’

where the kernel s Application

7-l

(X

I xa(xz” I 7-l I &xx;%’ I 7 I w41

17

(31)

is defined by / 7-l

1X”)(X”

17 1x’) dx” = (x 1x’) = 6(x - x’).

(32)

of the 8 operators to this yields

A?(+)(37 L?“-‘(3)

/ dx,” dx;

x [-(XI’XZ’ I D’P’W’P’

I 4x&x; I 7-l I -$‘@z”x,’ I DpXDp I XAJI

= I dx; dx,” h’xz’ I D’/OWp X (xz” 17-l 1xi’&’

1P’v’)(P’v’

I 6(x;> I PV>(PV I 4 I #(x2”> 1QW’P

(33)

I XI%),

the sign change being due to reordering of x; and x3’. Comparing hand side of Eq. (29) shows

with the right-

(D’P’P’v I Sx I ~vDt.4 =-

I

dx,” dx,” @‘CL’ I 6(x2 I PV>(-G I 7-l I G’XP’V’ I ?&‘I I Q.4

(34)

198

REMLER

AND

MILLER

Before going on with the reduction of this term to a more useful form we proceed with the analogous derivation of the single scattering contribution which arises from the expression

Application

of the limit operators to this and comparison of terms as before yields

= I dy, @a d.s dyl’ dvz’ 4

@‘CL I Y;Y;)
17’ 1YsYa)(Ys 1Pv)(y,y,

x (YI’ 1 7-l 1Y&Ys’Yz’

I ys’> 1Q-4.

(36)

It should be evident that further terms in the multiple scattering series can be obtained with similar ease, elegance and similarity to nonrelativistic procedures. In momentum space these formulas read,

I $RW * Y - W IP.&

TX@‘; E) = -CW<&tcp

x (N ’ Y - M)-‘
where the exchanged momentum

I (N ’ Y - M) $6)) I &PE)

(37)

N is

N=Dp-pE=DE-pF

(38)

and T,(e

E) = (24-l X

WF

I dk (DF/+ -

k

I ijDF - k) ii(ppvF)(k

PF ;PE,DE

-

k)u(p&(-B&

. y - M) + klDep~>,

(39)

where we use the symbols S(P1 + P2 - D)(+(PI - PZ>I Q-4

zz (~1~2 I DP> = s dx, dxz(PI I ~J(Pz I ~2Kw2 &PI + ~2 - DW/J

I D/G

(40)

I (1/2)(~1 - PZ)) =
Evaluation of single scattering as given by Eq. (39) requires knowing the Bethe-Salpeter amplitude with both nucleons off mass shell. Each amplitude contains a (k * y - M)-l factor so that the integrand contains a (kB - AI*)” pole.

NUCLEON-DEUTERON

ELASTIC

199

SCATTERING

Following the suggestion of an earlier analysis [ 171 of the deuteron’s electromagnetic form factor which involves a similar integral, we now approximate by retaining only the nucleon pole contribution. In effect one writes 277i(k - y - M)-l dk M - 2 u(kv) zZ(kv)(M/k”) Y

dk.

(41)

Such an approximation is of a similar category to neglect of resonance contributions mentioned in the introduction. Utilizing the relations 3/2 Ei(kv) lim

I D/4 =

W+-

(4 I &O>I kv) =

@7~)-~‘~

[(r . k - M)(&D

- k 1Dp)],

(41)

k%M=

,jii$ Wp I -W - My * k - M)14W,

(42)

Eq. (39) can be rewritten as

G(J’i E> =

@-F3(%+

I $

x &w&kv

I kv)

~~(PFVF)

TU’,

-

k,

PF

;

PE , DE - k)

I @9 I DEPE>.

(43)

Summation and integration over the repeated indices k, v is understood. Summation over Dirac spinor indices is of course implied and can be kept straight by following momentum variables. Evaluation of Eq. (43) requires knowledge of T(DF - k, pE ; pF , DE - k) at points where (DF - k)2 # M2, (DE - k)2 # M2. A prescription is required to relate values off-shell to experimentally determined on-shell values. T is a 16 x 16 matrix function of the invariant variables formed out of the four-vectors in its argument list. Since pE and pF are already on shell, there are four independent variables which may be taken to be t =

(PF

-

pd2,

ME2 = (DE - k)2,

u = (DF - k - PE)~; &IF2 = (DF - k)2.

(44)

Our prescription consists in neglecting the dependence of Ton the mass variables. This prescription was originally suggested by J. Wilson (private communication). Explicitly, it is the assumption T(t, U, f%fE2,hfF2) M T(t, U, M2, hf’).

(45)

This statement is exactly correct for the one boson exchange contributions to T and corresponds to preserving dependence on nearest singularities. It is just such

200

REMLER

AND

MILLER

most rapidly changing pieces of the nucleon-nucleon amplitude which, roughly speaking, should be most pertinent to our attempt here to calculate the most rapidly changing pieces of the nucleon-deuteron amplitude. We finally might note that any such mass dependence should be interrelated to isobar contributions already neglected. One would be ready at this point to calculate TX and Tl were it not for the fact that (kv 1#(O) 10~) decomposes into the sum of four invariant functions times their respective spinoral amplitudes whereas the nonrelativistic deuteron wavefunction, our present sole source of information concerning this amplitude, gives information concerning only two invariant functions corresponding to the S- and D-state. This question has been discussed in detail in a previous paper [12]. Two of the four invariant amplitudes have been singled out and with justification given, have been identified with the usual nonrelativistically obtained 4 and D-state wavefunctions. The contributions of the remaining two invariant functions to TX and Tl are kinematically suppressed at low energies. Since we have no information about them on hand at present we have ignored them here. Their status as sources of future corrections is, however, similar to that of the isobar contributions. The net result of these considerations is that one can write (kv 1 $(O) 1Dp)

= -(2~)-~/~

c u(D v’v”

k v’) n(p) . @(D

- k v’; kvH)(iu2)y~y

+ negative energy parts,

(46)

where the “negative energy parts” are what is ignored, uz is the usual Pauli matrix, D - k denotes an on-shell four-vector obtained from D - k in a manner we will

discuss shortly and finally, zJ(+) is a four-vector given by Ff’(P2V2 ; PlVl) = (4TF2]%

Y,(P 2v2 ; PPl) -

F’zY2(PV22

; PI4

(47)

q~,,,~are the usual momentum space deuteron wavefunctions dependent only on one invariant variable. Y,,, are essentially spin-spherical harmonics which reduce to the usual ones in the deuteron’s rest frame but must be relativistically transformed using Wigner rotation matrices to be evaluated in other frames. The detailed formulas are completely covered in [ 12 and 131. The definition of D - k entails a minor subtlety having potentialities of confusion not previously discussed. The origin of the separation of this deuteron vertex into positive and negative energy parts lies in a corresponding decomposition of (r * (D - k) + M)/2M which appears in expressions involving this vertex. An operator (y . N + M)/2M in which in general Na # M2 can be decomposed into a positive and negative energy projection operator by the following procedures.

NUCLEON-DEUTERON

ELASTIC

SCATTERING

201

(i) Pick an arbitrary reference frame. (ii) In this frame, keeping the spatial parts of N fixed, change its timelike component so that it goes on shell and thus define m = (AO, N)

(48)

J/o E gu2 + My.

(iii) Now one can write (y . N + M)/2M

= (l/2)(1 + N”/JP)(y . m + M)/2M + (l/2)(1 - NO/NO)(-y * P + M)/2M,

(49)

where P = (No, -N).

(50)

This is a decomposition into a sum of a positive and negative energy projection operators. These equations can be written in a formally covariant form by utilizing any timelike four-vector which defines the reference frame of choice. The possible confusion arises from one’s freedom of choice of this four-vector (i.e., frame). In scattering processes involving deuterons at least three natural frames present themselves each having certain advantages and disadvantages. The one used in our investigations may be called the D-frame, that in which the deuteron is at rest. For elastic scattering the entering and final D-frames differ for nonforward scattering. A second possibility may be called the C-frame, i.e., the center-of-mass frame for the particular process under consideration (i.e., it thus is process dependent). Another possibility may be called the N-frame, that in which the off-shell nucleon is at rest. This has one obvious disadvantage in that it is undefined for spacelike N. Yet another intriguing possibility would be to choose some infinite momentum frame. Given our present level of sophistication, a choice between such possibilities rest on one’s taste. Our choice of the D-frame defines D - k and therefore the wavefunction t,U+). Henceforth we drop the (+) superscript and understand that its momentum arguments lie on mass shell. Further useful properties are NPlQ

; P2V2)

=

#*(P2v2

; Pl"3

=

c

(i~2)“zY;

Yl’Y8’

~(p2v2’;pll’l’)(i~2)“~~yl.

(51)

Inserting these results into our previous expressions for TX and Tl yields the final working formula’s: TX(PFVF , = X

DF~F;PEVE~

D.r+.d

2 4PE) * #(Pm v)yvp~ #+(PE%;

NFVF')

; NEvE’) U(NEvE’)(y

* r*@F),

. N - iW)-l u(NFv/)

(52)

202

REMLER AND MILLER

- + vE;F, F - kvF’) ; j-dk(4k”)u@----X ~(PEVB) WE x @(DF - k

vF';

k

~E')[~(PE)

*

~~PF~F)WF

-

kp~

;PF,DE

-

k)

#(DE - k VE’; kv)]

kv) . r*(pF),

where N E.F

=

N

-

6’ = (M2 - N2)/D&,

~DE.F;

NE2 = NF2 = M2

and DE,F - k refer to D,,,-frames, respectively. These formula’s reduce of course to those obtained from nonrelativistic potential theory in the low energy limit. What we have here is then a relativistic extension of such formula’s which we believe makes maximal practical use of what little basic strong interaction theory is presently known. A report on the results of numerical computations based on these formula’s forms the bulk of the remaining sections of this paper.

III.

FITS TO EXPERIMENT

Table I is a listing of the experimental points used in this analysis along with the theoretical values obtained in the fitting procedure. An energy independent fit used the preliminary model specified by Eq. (2). Observables calculated using the resultant TX+,+, will be labeled X + 1 + S. Similarly there are results labeled X + 1, X, 1 etc., which are calculated using the corresponding contributions to the transition matrix. The full model discussed in the introduction requiring the calculation of double scattering also, can thus be labeled X + 1 + 2 + S. It was not obvious a priori at what energy, if any, the X + 1 + S procedure would be reasonable. Our low energy limit of 31 MeV was chosen so that there might be overlap with previous analyses of n-d elastic scattering which have been done mainly at lower energies. With respect to the high energy limit, preliminary calculations showed that the X + 1 + S model cannot possibly account for the backward peak seen in recent 316 MeV data. In fact it seems highly unlikely that any treatment which neglects the existence of nucleon isobars can be valid near or above this energy. 198 MeV was the next highest energy for which a recent and relatively complete data set was available.

NUCLEON-DEUTERON

ELASTIC TABLE

Experiments

Rim et al. [20] 31 MeV p-d cross sectiot?

a Points b “Cross

enclosed section”

I

and Theoretical

c.m. Anglea (deg.)

Identification

(7.6) (9.1) (10.6) (13.6) (15.1) (16.6) (18.1) (21.1) (24.1) (27.1) 30.1 37.5 44.8 52.0 59.1 66.1 72.9 79.6 86.1 92.4 98.5 104. 110. 115. 120. 125. 130. 134. 138. 142. 146. 149. 153. 156. 159. 162. 165. 167. 170.

203

SCATTERING

Results

Experimental value”

Experimental uncertainty*

Theoretical valud

107. 66.8 58.4 53.7 53.8 57.3 60.9 60.3 63.9 60.8 62.0 49.8 41.3 33.9 26.9 21.4 17.0 13.8 11.8 9.01 7.77 5.96 4.97 3.80 2.85 2.38 2.29 2.60 3.64 5.05 7.58 10.5 14.6 19.1 24.9 28.0 32.2 35.9 39.4

11.0 5.3 4.10 3.50 2.20 2.80 2.40 1.70 1.70 1.60 1.70 1.30 0.900 0.800 0.600 0.600 0.400 0.500 0.400 0.300 0.270 0.210 0.170 0.150 0.120 0.110 0.100 0.110 0.150 0.230 0.300 0.400 0.500 0.600 1.00 0.800 1 .oo 1 .oo 1.10

73.7 72.6 71.6 69.3 68.0 66.8 65.6 62.9 60.1 57.3 54.3 46.8 39.1 29.7 24.3 20.0 16.4 13.2 10.8 8.76 7.09 5.64 4.42 3.36 2.55 2.15 2.22 2.77 3.93 5.63 8.12 11.3 14.9 19.1 23.5 28.1 32.5 36.7 40.5

in parentheses were not used in fitting. means cm. differential cross section.

Cross-section

values

are in mbisr.

204

RJIMLER

TABLE Ashby [21] 31 MeV p-d

cross

section

Romero et al. [22] 36 MeV n-d cross section

AND

MILLER

I (conrinued)

(22.4) (22.4) 33.5 33.5 44.5 44.5 55.2 55.2 55.2 60.0 65.7 65.7 75.0 75.9 85.7 90.0 90.0 90.0 95.0 105. 105. 105. 112. 120. 120. 135. 135. 150. 150.

58.5 57.6 49.6 47.3 37.6 36.4 29.3 29.7 31.8 23.5 19.2 19.0 14.5 12.7 9.86 8.02 8.14 10.2 6.55 5.49 5.05 4.84 2.33 2.93 2.70 2.20 2.10 8.82 9.20

1.58 1.50 1.49 1.23 0.790 0.764 1.08 0.742 0.700 2.21 0.518 0.532 0.406 0.432 0.394 0.289 0.285 0.428 0.426 0.253 0.217 0.194 0.161 0.123 0.173 0.183 0.227 0.388 0.396

61.7 61.7 50.9 50.9 39.4 39.3 27.2 27.2 27.2 23.7 20.2 20.2 15.2 14.8 10.9 9.54 9.54 9.52 8.07 5.49 5.48 5.48 3.87 2.67 2.67 3.00 2.97 11.9 11.9

(15.1) (22.6) 30.1 37.5 44.8 59.2 73.0 84.2 86.2 89.2 94.3 99.3 98.6 104. 110. 110.

95.7 62.0 56.0 40.8 32.1 21.5 11.6 8.10 6.30 8.40 6.10 5.50 5.80 3.20 3.20 3.10

7.70 5.00 4.20 3.30 2.50 1.70 0.900 0.700 0.500 1.00 0.500 0.500 0.600 0.300 0.200 0.300

60.9 54.3 47.2 39.9 32.6 19.0 12.4 8.64 8.11 7.34 6.19 5.17 5.30 4.22 3.31 3.38

NUCLEON-DEUTERON

TABLE Romero et al. [22] (continued)

Bunker et al. 35 MeV p-d

[23] cross

section

ELASTIC

205

SCATTERING

I (continued)

120. 121. 125. 130. 133. 140. 145. 150. 153. 160. 170.

1.60 1.70 1.30 1.40 1.10 2.40 3.60 7.60 6.70 14.8 20.1

0.100 0.300 0.100 0.200 0.100 0.200 0.300 0.600 0.400 0.900 1.70

2.02 1.92 1.57 1.54 1.69 2.98 5.01 8.13 10.7 18.3 30.4

(33.1) (36.0) (39.0) (41.9) (44.8) (47.7) (50.6) (53.5) (56.3) (59.2) (62.0) (64.8) (67.5) (70.3) (73.0) (75.7) (78.3) (81.0) (83.6) (86.1) (88.7) (91.2) (93.7) (96.1) (98.5) (101.) (103.) (105.) (108.) (110.) (112.) (114.) (116.) (118.)

45.0 41.9 38.4 35.1 32.1 29.2 26.5 24.0 21.6 19.5 17.5 15.8 14.8 12.8 11.6 10.5 9.61 8.76 7.86 7.27 6.61 5.97 5.52 4.84 4.51 4.32 3.98 3.67 3.37 3.08 2.79 2.54 2.28 2.21

0.360 0.502 0.384 0.386 0.353 0.350 0.345 0.336 0.324 0.293 0.262 0.237 0.214 0.192 0.175 0.157 0.144 0.131 0.118 0.102 0.099 0.096 0.088 0.083 0.045 0.043 0.052 0.055 0.051 0.049 0.050 0.041 0.030 0.031

44.2 41.3 38.3 35.4 32.5 27.1 24.8 22.6 20.7 18.9 17.3 15.9 14.5 13.3 12.2 11.1 10.2 9.38 8.62 7.93 7.28 6.68 6.13 5.61 5.12 4.67 4.24 3.87 3.49 3.13 2.81 2.51 2.25 2.02

206

REMLER

TABLE

et al. [23] (continued)

AND

MILLER

I (confinued)

Bunker

(120.) (120.) (122.) (1W (124.) (128.) (132.) (136.) (140.) uJJ.1 (148.) (152.) (156.) (160.) (1W (168.)

1.93 2.02 1.75 1.66 1.78 1.52 1.54 1.76 2.29 3.54 5.13 7.78 11.0 14.3 18.8 22.9

0.095 0.039 0.016 0.066 0.105 0.038 0.019 0.056 0.142 0.273 0.380 0.498 0.561 0.686 0.827 1.10

Bunker et al. [23] 35 MeV p-d polarization

(26.5) (30.0) (37.5) (41.8) (44.9) (48.5) (52.1) (59.1) (66.1) (73.1) (78.9) (86.1) (92.4) (98.5) (1W (110.) (115.) (120.) (125.) (130.) (134.) ww (145.) (150.) (155.) (159.) WV

0.053 0.067 0.066 0.052 0.089 0.086 0.082 0.051 0.020 -0.002 -0.055 -0.127 -0.175 -0.217 -0.273 -0.313 -0.312 -0.291 -0.260 -0.035 0.079 0.199 0.198 0,168 0.120 0.084 0.053

0.012 0.006 0.009 0.009 0.011 0.011 0.007 0.008 0.009 0.026 0.013 0.038 0.044 0.051 0.056 0.057 0.060 0.050 0.051 0.047 0.041 0.047 0.030 0.032 0.024 0.017 0.015

1.89 1.83 1.69 1.61 1.59 1.54 1.77 2.34 3.45 5.19 7.68 11.0 15.0 20.2 25.7 30.8

0.155 0.172 0.206 0.223 0.234 0.243 0.252 0.268 0.274 0.269 0.246 0.224 0.200 0.175 0.148 0.126 0.094 1.048 -0.011 -0.060 -0.073 -0.050 -0.034 -0.021 -0.013 -0.008 -0.005

NUCLEON-DEUTERON

ELASTIC

TABLE

Williams and Brussel 1241 40 MeV p-d cross section

595lW-14

I

(6.0) (7.5) (9.0) (10.5) (12.0) (13.5) (15.0) (16.5) (18.0) (19.5) (21.0) (22.4) (23.9) (25.4) (26.9) (29.9) 37.2 40.9 44.5 51.7 58.8 65.7 72.6 79.2 85.7 92.0 98.1 104. 110. 120. 140. 146. 148. 150. 152. 154. 158. 158. 160. 162. 164. 166. 168. 170. 172.

207

SCATTERING

(continued) 14.5. 60.1 42.8 41.5 41.6 44.8 46.7 49.1 49.2 48.9 48.7 48.3 46.6 47.0 45.2 43.2 35.3 31.5 27.6 23.0 17.1 13.4 9.65 8.43 6.12 5.20 4.29 3.90 3.61 2.19 2.89 3.61 4.43 5.03 6.08 6.74 8.38 10.3 12.5 13.8 15.5 16.0 18.9 19.3 20.9

1.80 0.890 0.890 0.890 0.890 0.900 0.900 0.900 0.450 0.910 0.900 0.460 0.460 0.920 0.920 1.39 0.950 0.970 0.490 0.510 0.530 0.390 0.240 0.250 0.200 0.150 0.160 0.170 0.280 0.810 0.140 0.130 0.900 0.120 0.180 0.180 0.150 0.210 0.250 0.260 0.270 0.310 0.360 0.370 0.420

62.1 60.8 59.7 58.5 57.3 56.0 54.9 53.6 52.3 51.0 49.7 48.4 47.1 45.8 44.5 41.8 35.1 31.8 28.6 20.8 16.7 13.7 11.3 9.39 7.87 6.64 5.58 4.64 3.77 2.33 2.06 3.63 4.39 5.35 6.66 7.91 11.0 11.1 12.9 14.7 16.7 18.4 20.6 22.2 23.9

208

REMLER

TABLE

AND

MILLER

I (conrhuecf)

Conzett et al. [25] 0 MeV p-d polarization

(30.1) (37.5) (44.9) (52.1) (57.9) (66.2) (73.0) (79.7) (87.7) (92.5) (98.6) (low (106.) (110.) (114.) (120.) (125.) (130.) (135.) (142.)

0.093 0.123 0.103 0.128 0.098 0.040 -0.012 -0.095 -0.250 -0.294 -0.281 -0.327 -0.331 -0.374 -0.392 -0.385 -0.310 -0.366 0.060 0.490

0.008 0.012 0.011 0.012 0.010 0.016 0.017 0.016 0.022 0.018 0.015 0.061 0.023 0.033 0.018 0.045 0.030 0.036 0.041 0.200

0.187 0.220 0.245 0.253 0.257 0.247 0.227 0.185 0.141 0.113 0.073 0.063 0.039 0.010 -0.025 -0.077 -0.141 -0.187 -0.187 -0.121

Bunker et al. [23] 40 MeV p-d polarization

(120.) (125.) (130.) (135.) (140.) (145.) (150.) (155.) (160.) (164.)

- 0.420 -0.299 -0.183 0,061 0.186 0.213 0.169 0.127 0.051 0.057

0.020 0.023 0.046 0.033 0.026 0.019 0.014 0.015 0.024 0.023

-0.076 -0.134 -0.185 -0.198 -0.140 -0.083 -0.042 -0.022 -0.011 -0.007

70.7 52.2 41.9 32.0 23.4 15.4 14.5 9.30 9.80 6.40 7.40 5.00 3.80 3.60

5.80 4.20 3.50 2.80 1.80 1.30 1.10 0.900 0.800 0.500 0.500 0.500 0.400 0.200

49.0 37.9 35.7 29.1 23.0 15.8 12.3 9.64 8.62 7.71 6.25 6.23 5.09 4.56

Romero et al. [22] 46 MeV n-d cross

section

(15.2) (27.7) 30.2 37.6 44.9 52.2 59.3 66.3 69.7 73.1 79.7 79.8 86.3 89.7

NUCLEON-DEUTERON

ELASTIC

TABLE Romero

et al. [22]

(continued)

Bunker

et al. [23]

209

SCATTERING

I (continued)

98.7

2.30

0.300

99.7 110. 120. 130. 140. 150. 160. 170.

2.60 1.90 1.40 0.880 1.50 3.00 6.60 11.9

0.200 0.200 0.200 0.160 0.200 0.200 0.400 0.800

3.44 3.32 2.31 1.53 1.26 2.08 5.26 11.8 20.0

(9.1) (12.1) (15.1) (18.2) (21.2)

36.4 38.2 40.6 43.8 42.7 41.2 38.7 34.1 31.2 28.5 25.5 23.0 20.5 18.3 16.4 13.7 11.6 10.6 9.08 7.74 6.55 6.08 5.25 4.57 4.05 3.83 3.49 3.13 2.80 2.69 2.39 2.17 2.00 1.91 1.64 1.47

1.06 1.07 0.853 0.525 0.256 0.288 0.348 0.409 0.281 0.228 0.229 0.299 0.267 0.238 0.164 0.137 0.151 0.127 0.082 0.093 0.098 0.073 0.047 0.046 0.032 0.031 0.038 0.031 0.028 0.030 0.029 0.024 0.022 0.019 0.053 0.032

54.4 51.8 49.1 46.5 43.8 41.1 38.4 35.7 33.1 30.5 27.9 25.4 23.0 18.5 17.1 14.6 12.5 11.7 10.3 8.96 7.87 7.37 6.62 5.84 5.18 4.88 4.43 3.94 3.50 3.30 2.98 2.63 2.31 2.16 1.80 1.52

(24.2) (27.2) (30.2) (33.1) (36.1) (39.1) (42.0) (44.9) (47.8) (50.1) (54.4) (58.7) (60.8) (64.3) (68.5) (72.5) (74.5) (77.9) (81.8) (85.7) (87.6) (90.8) (94.5) (98.1) (99.9) (103.) (106.) (110.) (111.) (116.) (120.)

210

REMLER

TABLE

et al. [23] (continued)

Bunker

Johnston 49 MeV

Davison 77 MeV

et al. [26] p-d polarization

et al. [27] p-d

cross

section

AND

I

MILLER

(continued)

(124.) (128.) (132.) (136.) (140.) (144.) (148.) (152.) (156.) (160.) (162.) (164.1 (166.) (168.)

1.39 1.31 1.28 1.37 1.74 2.40 3.35 4.67 6.39 8.14 9.46 10.7 12.2 13.4

0.026 0.022 0.017 0.022 0.063 0.101 0.134 0.168 0.192 0.252 0.312 0.375 0.462 0.588

1.33 1.24 1.31 1.54 2.10 3.04 4.44 6.37 8.85 11.8 13.5 15.2 16.7 18.5

(112.) (120.) (125.) (130.) (135.) (140.) (145.) (150.) (155.) (159.) (163.) (165.) (168.) (170.) (173.) (175.)

-0.483 -0.492 -0.401 -0.200 -0.028 0.105 0.185 0.151 0.141 0.116 0.095 0.064 0.063 0.050 0.041 0.009

0.027 0.022 0.022 0.024 0.021 0.026 0.015 0.016 0.012 0.011 0.012 0.012 0.012 0.011 0.011 0.011

0.078 0.016 -0.029 -0.068 -0.076 -0.064 -0.042 -0.023 -0.012 -0.007 -0.004 -0.002 -0.002 -0.001 -o.ooo -o.ooo

197. 28.3 36.9 23.4 16.5 9.75 4.34 3.19 1.92 1.31 0.891 0.916 0.976 0.675

26.7 5.76 4.07 2.51 1.67 1.02 0.667 0.619 0.496 0.349 0.292 0.281 0.305 0.163

35.6 33.1 29.1 22.6 15.9 10.3 5.19 3.38 2.38 1.73 1.29 1.00 0.743 0.588

(8.9) (11.4) (16.0) (24.4) 34.2 44.3 54.8 64.7 74.9 84.6 95.0 104. 115. 125.

NUCLEON-DEUTERON

ELASTIC

TABLE

et al. [27] (continued)

Davison

(continued)

135. 145. 155. 165. 175.

Chamberlain and Stern 190 MeV d-p (= 95 MeV p-d) cross section

[28]

Poulet et al. [29] 138 MeV p-d polarization

Postma and Wilsond 146 MeV p-d cross section

I

[30]

c We have not done d Experimental cross

211

SCATTERING

0.556 1.10 1.49 2.73 6.18

0.255 0.393 0.551 0.898 2.53

0.673 1.24 2.84 5.53 8.30

(15.4) 31.1 38.4 48.1 57.8 67.6 77.5 81.5 87.5 97.5 108. 118. 128. 138. 149. 159. 170.

31.1 8.90 5.30 4.00 2.33 1.22 1.16 0.890 0.710 0.640 0.610 0.520 0.550 0.400 0.610 1.52 1.75

5.10 1.30 0.500 0.400 0.210 0.100 0.100 0.080 0.060 0.050 0.050 0.090 0.040 0.050 0.060 0.200 0.340

23.2 13.1 9.51 4.72 2.85 1.91 1.34 1.18 0.982 0.747 0.564 0.442 0.401 0.545 1.10 2.44 4.33

(23.1) (30.7) (45.7) (60.3) (81.0) (99.0) (119.2) (139.4) (149.5) (159.7)

.476 .576 .349 .054 -446 -.680 -.410 .054 .172 .135

(3.9) (4.7) (6.3) (7.7) (9.3) (10.9) (12.4) (13.9)

the single scattering integral sections have been multiplied

115. 42.6 17.9 14.2 17.9 18.4 16.1 16.6 at this energy. by 0.92 (see [32]).

.037 .050 .043 .057 .056 .170 .045 .034 .041 .035 10.5 3.86 1.63 1.29 1.62 1.66 1.45 1.51

c

22.8 21.9 20.3 18.9 17.5 16.2 15.1 14.0

212

REMLER TABLE

AND

MILLER

I (conhued)

[30]

(15.5) (19.3) (23.1) 30.8 38.4 45.8 53.2 60.4 67.5 75.1 81.3 87.6 93.9 99.0 106. 109. 120. 130. 140. 150. 155. 160. 165. 170.

16.5 11.9 10.4 5.40 3.64 2.36 1.70 1.14 0.906 0.584 0.492 0.390 0.335 0.263 0.263 0.245 0.235 0.247 0.280 0.326 0.406 0.477 0.582 0.795

1.49 1.67 1.62 0.460 0.360 0.230 0.170 0.120 0.119 0.047 0.049 0.039 0.034 0.042 0.026 0.040 0.034 0.030 0.031 0.033 0.050 0.038 0.068 0.097

13.0 10.8 9.08 6.42 4.55 2.38 1.57 1.15 0.920 0.649 0.522 0.425 0.354 0.309 0.260 0.240 0.196 0.186 0.230 0.375 0.520 0.727 0.995 1.26

Postma and Wilson [30] 146 MeV p-d polarization

3.9 4.7 6.3 7.7 9.3 10.9 12.4 13.9 15.5 19.3 23.1 30.8 38.4 45.8 53.2 60.4 67.5 75.1 81.3 87.6 93.9

0.083 0.207 0.287 0.252 0.290 0.309 0.332 0.369 0.406 0.428 0.550 0.681 0.676 0.512 0.278 0.075 -0.179 -0.299 -0.416 -0.429 -0.497

0.058 0.039 0.027 0.026 0.026 0.026 0.026 0.026 0.026 0.063 0.057 0.034 0.031 0.030 0.036 0.046 0.053 0.047 0.050 0.049 0.047

0.080 0.098 0.133 0.165 0.201 0.237 0.272 0.308 0.347 0.428 0.501 0.602 0.614 0.526 0.355 0.111 -0.119 -0.128 -0.233 -0.331 -0.407

Postma

and Wilson

(continued)

NUCLEON-DEUTERON

ELASTIC

TABLE Postma and Wilson (continued)

[30]

Igo et al. 1311 291 MeV d-p (= 146 MeV p-d) cross section

Palmieri [32] 152 MeV p-d

cross

section

Kuroda et al. [33] 155 MeV p-d cross

section

213

SCATTERING

I (continued)

99.0 106. 109. 120. 130. 140. 150. 155. 160. 165. 170.

-0.639 -0.623 -0.592 -0.488 -0.044 0.116 0.178 0.235 0.173 0.090 0.112

0.061 0.056 0.098 0.105 0.067 0.042 0.051 0.058 0.047 0.056 0.056

-0.452 -0.443 -0.450 -0.406 -0.240 -0.014 0.117 0.122 0.104 0.077 0.050

(114.) (117.) (120.) (124.) (127.) (127.) (130.) (130.) (133.) (137.) t1w (140.) (143.) (143.) (146.) (150.) (153.) (157.)

0.197 0.186 0.204 0.214 0.208 0.219 0.227 0.224 0.233 0.248 0.231 0.270 0.251 0.272 0.282 0.316 0.358 0.383

0.010 0.009 0.010 0.010 0.014 0.011 0.014 0.012 0.026 0.014 0.013 0.017 0.014 0.018 0.015 0.016 0.018 0.019

0.217 0.204 0.195 0.188 0.185 0.185 0.186 0.187 0.192 0.210 0.227 0.231 0.258 0.263 0.304 0.370 0.461 0.580

(99.0) (109.) (119.) (129.) (139.) (150.) (155.) (160.) (165.) (170.)

0.241 0.184 0.183 0.214 0.291 0.340 0.403 0.558 0.657 0.691

0.018 0.021 0.015 0.017 0.017 0.024 0.027 0.036 0.036 0.043

0.285 0.233 0.204 0.197 0.225 0.322 0.427 0.582 0.788 1.00

(4.5) (7.5) (10.0) (15.1)

24.7 13.9 14.5 14.3

3.00 1.00 1.00 1.00

21.5 18.5 16.3 12.6

214

REMLER

TABLE

AND

MILLER

I (continued)

(19.0) (23.0) (28.5) 38.5 46.0 53.0 60.0 67.5 82.0 89.0 99.0 109. 120. 130. 143. 150. 155. 160. 170.

11.5 9.20 6.55 3.28 2.02 1.30 0.930 0.770 0.470 0.320 0.246 0.215 0.214 0.236 0.283 0.316 0.370 0.478 0.770

0.800 0.650 0.400 0.200 0.140 0.080 0.060 0.050 0.030 0.025 0.020 0.015 0.013 0.013 0.017 0.012 0.015 0,026 0.050

10.4 8.64 6.71 4.30 2.21 1.47 1.06 0.809 0.453 0.366 0.285 0.233 0.204 0.197 0.239 0.311 0.404 0.550 0.940

Kuroda et al. [33 ] 155 MeV p-d polarization

7.5 11.5 15.5 19.0 23.0 28.5 34.0 38.5 46.0 53.0 64.0 82.0

0.329 0.358 0.341 0.473 0.494 0.557 0.496 0.526 0.351 0.243 -0.057 -0.297

0.030 0.024 0.018 0.020 0.024 0.025 0.032 0.029 0.037 0.035 0.042 0.100

0.162 0.256 0.350 0.430 0.506 0.585 0.617 0.605 0.506 0.334 -0.046 -0.259

Adelberger 198 MeV

80.0 85.0 90.0 92.5 95.0 100. 105. 105. 105. 110. 115. 120. 120.

0.197 0.165 0.138 0.131 0.121 0.108 0.103 0.097 0.106 0.106 0.097 0.104 0.104

0.010 0.007 0.005 0.004 0.004 0.004 0.003 0.003 0.004 0.003 0.003 0.003 0.003

0.229 0.183 0.151 0.138 0.128 0.112 0.102 0.102 0.101 0.094 0.091 0.093 0.093

Kuroda et al. 1331 (continued)

and Brown 1341 p-d cross section

NUCLEON-DEUTERON TABLE

Adelberger

and Brown

[34]

(continued)

Adelberger 198 MeV

and Brown [341 p-d polarization

125. 129. 135. 138. 139. 139. 145. 149. 150. 155. 160. 165. 80.0 85.0 90.0 92.5 95.0 loo. 105. 105. 105. 109. 110. 115, 120. 125. 129. 129. 135. 138. 139. 139. 145. 149. 150. 150. 155. 160. 160. 165. 170.

ELASTIC I

215

SCATTERING

(continued) 0.111 0.123 0.148 0.155 0.169 0.157 0.186 0.204 0.214 0.242 0.290 0.346 -0.480 -0.493 -0.504 -0.541 -0.478 -0.433 -0.399 -0.436 -0.416 -0.287 -0.300 -0.254 -0.139 -0.078 0.021 -0.010 0.091 0.138 0.105 0.120 0.197 0.214 0.204 0.200 0.178 0.145 0.167 0.153 0.048

0.004 0.004 0.005 0.006 0.005 0.005 0.006 0.009 0.007 0.008 0.011 0.015

0.099 0.107 0.126 0.139 0.141 0.141 0.180 0.212 0.222 0.281 0.361 0.457

0.010 0.006 0.005 0.010 0.005 0.005 0.008 0.009 0.010 0.030 0.009 0.011 0.013 0.009 0.025 0.008 0.009 0.013 0.009 0.008 0.009 0.035 0.012 0.009 0.009 0.025 0.010 0.015 0.025

-0.317 -0.399 -0.459 -0.478 -0.488 -0.486 -0.441 -0.441 -0.441 -0.404 -0.393 -0.321 -0.230 -0.132 -0.059 -0.058 0.022 0.055 0.060 0.060 0.105 0.117 0.117 0.117 0.111 0.095 0.095 0.070 0.045

216

REMLER

AND

MILLER

The fits at 31,35,40, and 46 MeV are similar both with respect to procedure and results. At each energy independently, (54) was minimized with respect to variations in D and Q (see Eq. (2)). a(B), uT denote the differential and total cross sections respectively. a&, was obtained by interpolating a tabulation given in [40]. The fits are shown in Fig. 2. In all these cases we do well in cross-section except at large backward angles where theory is at times high by factors of up to 50 %.

0

‘!30 ‘I!

60

”

! ” 120! ” 150! “I

90

F Y AUCl F

I80

FIG. 2. Center-of-mass differential cross section at low energies. The solid lines are our fits. The data are from Refs. [20, 21, 22, and 241.

Errors of this order of magnitude are consistent with what might be expected from the missing p-wave component of double scattering. At forward angles where single scattering is very large an error in one partial wave is relatively unimportant. The backward exchange peak is however considerably smaller and so the P-wave error has relatively larger effect. Figure 3 shows the differential cross section due to various pieces of the TX+,+, matrix at 46 MeV. These results are typical of this energy group. We note that the

NUCLEON-DEUTERON

ELASTIC

SCATTERING

217

FIG. 3. Contributions to the theoretical differential cross section at 46 MeV. The long dashed line is single scattering (1). The short dashed line is nucleon exchange (X). The solid line is the full model, single plus exchangeplus S-wave (X + 1 + S). The sum of single and exchange(X + 1) is shown as a short solid line where it differs significantly from single or exchange.

effect of adding S-wave is to generally lower ax+1(0). At 180” the S-wave correction is a factor of 2. Similar but somewhat smaller destructive interference may be expected from the unaccounted for P-waves. The error to be expected in a theoretical prediction of polarization P is of order Au/u when Au is the error in differential cross section. At these low energies AU/U N P, except at forward angles so that our theoretical curves are not surprisingly quite poor. Figure 4 shows these theoretical predictions for polarization made on the basis of fits to the cross section and also shows experimental data at nearby energies. Inclusion of the polarization data in Eq. (54) was tried; this did not cause the position

of the minimum

in Q, D parameter

space to change significantly.

In

addition

extensive searches were made for other local minima with no success. One tactic however which did result in a dramatic improvement was the addition of certain P-wave terms to TX,,,, . This small step in the direction of a phase shift analysis will be described in some detail in Appendix

E. We mention here only the

218

REMLER

0.4

AND

MILLER

t

i

0.6

-

a E

0.4

--

35

MEV

40

MEV

2

-0.2 -0.4

y*

1

III

I 1I

FIG. 4. Low energy neutron polarization. The solid line is our fit. The data are from Refs. [23 and 251. The effect on these fits of a small additional P-wave is shown in Appendix E.

following major results relevant to this discussion. (i) Upon adding only two more search parameters, the real and imaginary parts of PLs in Tx+l+s+P’

=-

TX,,,

+PLS'IE*(PP)

*+-+d@~

x

PEIIPFI IPED * Q,

(55)

one can simultaneously fit cross section and polarization (qualitatively), (ii) the S-wave parameters’ minimum values are altered thereby by only a few percent. This second result is of particular importance to us here in that one might well doubt the meaningfullness of the Q and D values obtained using the X + 1 + S approach at these energies when the polarization values they imply are so wrong. We see, at least in this instance, that the different partial waves are for practical purposes, uncoupled from one another in the search procedure (much as the coefficients of orthogonal functions are independently determined in approximation theory). We therefore believe that the values of Q and D obtained by fitting using the X + 1 + S approach, realistically represent the S-wave components of the neglected parts of the scattering matrix. In the next section we give further evidence supporting this point.

NUCLEON-DEUTERON

-0.2

ELASTIC

SCATTERING

219

1

FIG. 5. Contributions to the theoretical neutron polarization at 46 MeV. The short dashed line is single scattering (1). The long dashed line is single scattering plus nucleon exchange (X + 1). The solid line is the full model, single plus exchange plus S-wave (X + 1 + S). Nucleon exchange by itself gives no polarization.

c

I

0

I

I

30

I

I

I

60

I

I,

I

I

90

I

120

I

I

I

150

I

I,

160

C. M. ANGLE

FIG. 6. Center-of-mass differential cross sections at middle energies. The solid lines are our fits. The data are from Refs. [27 and 281.

220

REMLER

AND

MILLER

Figure 5 shows the polarization due to various pieces of the TX+,+, matrix at 46 MeV (polarization due to exchange alone vanishes). The fits at 77 and 95 MeV are similar in that the data base is very small, and in particular there are no polarization measurements. These fits are shown in Fig. 6. The corresponding theoretical predictions of polarization are given in Fig. 7. 0.6 r 0.4 0.2

95 MEV

-0.2 -0.4

J-

7. Neutron polarizations at middle energies. The solid curves are onr fits. There is no polarization data in this region. FIG.

We now come to the final group of fits at 146, 155, and 198 MeV. Here a remarkable change has taken place and by a stroke of luck the X + 1 + S theory shows itself to be quite realistic. When the parameters Q, D were fitted to the cross section data alone, the results were again quite respectable but what is more important, the theoretical polarization predictions were qualitatively correct. We therefore fit at these energies the cross section and polarization data simultaneously, minimizing

at each energy independently

+ c [(P(O)- ~exrJ(~))/~~exp(~)12 + KJ - dA.J~~~x~1”. al1 0

(56)

30

C.M.

60

ANGLE

90

120

150

160

FIG. 8. Center-of-mass differential cross sections at higher energies. The solid lines are our fits. The data are from Refs. [30, 31, 32, 33, and 341. The Postma and Wilson data [30], has been multiplied by 0.92 (see [32]).

0

MEV

155

MEV

FIG. 9. Neutron polarization at higher energies. The solid lines are our fits. The data are from Refs. [30, 33, and 341.

-0.4

-0.2

0.0

0.2

0.4

0.6

$ 0.6 F 2 0.4 L 4 0.2 B 0.0 0' f -0.2 az -0.4

-0.4

-0.2

0.0

0.2

0.4

0.6

C.M.

ANGLE

FIG. 10. Contributions to the theoretical differential cross section at 198 MeV. The long dashed line is single scattering (1). The short dashed line is nucleon exchange (A’). The solid line is the full model, single plus exchange plus S-wave (X + 1 + S’). The sum of single and exchange (X + 1) is shown as a short solid line where it differs significantly from single or exchange.

lo-3lO

r

L

FIG. 11. Contributions to the theoretical neutron at 198 MeV. The short dashed line is single scattering dashed line is single scattering plus nucleon exchange solid line is the full model, single plus exchange G- + 1 + S).

-0.6

-0.4

0.6

polarization (1). The long (A’ + 1). The plus S-wave

NUCLEON-DEUTERON

ELASTIC

223

SCATTERING

The cross section and polarization results are shown in Figs. 8 and 9 respectively, along with very recent data [31, 321 at nearby energies received too late to be included in the fitting program. A breakdown of the various theoretical contributions at 198 MeV is shown in Figs. 10 and 11. These results show that at these energies in the region of the backward peak the pure exchange and single scattering terms are still much too high and that destructive interference provided by multiple scattering is still quite important. Thus any analysis of such data neglecting multiple scattering is most probably wrong. As mentioned before, at all these energies, the measured total cross sections were included as data points. Figure 12 shows experimental and theoretical values. For lab energies > 77 MeV theory is quite good and the S-wave addition improves somewhat the already good value of u:+~ . This we regard as further indication that the model is basically correct at these energies and that, since the forward amplitude is completely dominated by single scattering, that our theoretical computation of single scattering is realistic. Within the lower energy group of fits the discrepancy

50 30

I 40

I ,I,,, 50 60 80 LAB KINETIC

i loo ENERGY

I 200

I 300

(MEV)

FIG. 12. Nucleon-deuteron total elastic cross section. The data are from Ref. [40]. The dashed line is from the “Born” calculation, single scattering plus exchange (X + 1). The solid line is the full model, single plus exchange plus S-wave (X + 1 + s>. 595/g2/1-I5

224

REMLER AND MILLER

between theory and experiment is consistent in order of magnitude with those previously encountered. It is also interesting to note that there here the S-wave addition generally makes things worse. IV. PARTIAL WAVE ANALYSIS At each energy we have projected out the matrix elements p’+lL;

1(S,,,

- I)/(29 1-+lL,),

for all L, L’ < 2 and J < 5/2 by numerical integration. for example (%/2 I (S - mw

Our notation is such that,

I 2s1,2) = (exp(2i26,) - 1)/(2i),

(57)

where 2&, is the usual doublet S-wave phase shift. The results are given in Tables II-X. These numbers may be of interest, especially at lower energies, for phase shift analyses and for people utilizing Faddeev equation approaches. Figure 13 is the Argand diagram for (2s,,2 1 (S,,, - 1)/2i) I 2s1,J and (26~2I &+I+~ - WW I 2s,,2), the latter being obtained from the fits described in the previous section. Successive energy points are simply connected by straight lines. DOUBLET 1.0

+VAN OERS BROCKMAN

X*1 r

AND /

-0.5

I-

-I I.5

FIG. 13. Argand diagram for the doublet S-wave. The “Born” amplitude, single scattering plus nucleon exchange, is marked “X + 1”. The same amplitude with the addition of the S-wave from the fits is marked “X + 1 + S”. The phase shift values of van Oers and Brockman [18], which assume unsplit P-waves, are marked with a plus sign. The more recent Faddeev equation values of Pieper [7] are marked with an “X”. We use the values from his potential set “C” at 23 and 40 MeV and set “D” at 77 MeV. The small numbers are the laboratory kinetic energies in MeV.

NUCLEON-DEUTERON

ELASTIC

TABLE

225

SCATTERING

II

Partial Waves at 31 MeV” JP

zs+lL

1/2+

2s 4D

( 1.230, -0.040, 0.114,

5 2D 4D

(-0.364, 0.737y 0.484, 0.545 0.046, -0.023 0.132, 0.015

3/2+

l/23/25/2-

(Column labels, 2S+1L, same as row labels.) 1.009)” 0.741 0.033

-0.079,

0.060

0.154, 0.073 -0.014, -0.002

2P 4P

0.070, 0.018,

0.315 0.001

0.341,

0.244

=P 4P

0.035, 0.023,

0.315 0.000

0.494,

0.235

4P

0.405,

0.240

-0.089,0.059

GEach .7P submatrix is symmetric. Redundant elements are omitted. Imaginary parts follow real parts. See text for normalization conventions. L The numbers in parentheses are the S-wave components computed from the single scattering + exchange or X + 1 amplitudes. All other numbers are for the full, X + 1 + S, amplitudes. TABLE III Partial Waves at 35 MeV” Jp

zs+lL

1/2+

3/2f

l/23/25/2#JJSee Table II.

2s 4D 3 2D 4D *P 4P 2P 4P 4P

(Column labels, as+lL, same as row labels.) (1.144, 0.911y 0.052, 0.564 0.108, 0.031 (-0.283, 0.680)” 0.502, 0.471 0.048, -0.022 0.139, 0.017 0.085, 0.313 0.016, 0.002 0.047, 0.313 0.024, 0.001 0.396, 0.246

-0.068,

0.069

0.159, 0.080 -0.015, -0.002 0.338,

0.249

0.478,

0.241

-0.077, 0.066

226

REMLER

AND

TABLE Partial

os+lL

JP

1/z+

3/2+

l/2-

3/2-

s/2-

=vb See Table

Waves

(Column

3 4D

(1.049, -0.078, 0.098,

0.800)~ 0.406 0.028

3 %D 4D

(-0.203, 0.650, 0.051, 0.145,

0.621)b 0.373 -0.020 0.020

labels,

zs+lL, same as row labels.)

-0.055,

0.076

0.159, -0.015,

0.085 -0.001

0.099, 0.015,

0.305 0.004

0.338,

0.251

2P 4P

0.056,0.305 0.026, 0.001

0.459,

0.245

4P

0.387,

-0.064,0.073

0.248

II. TABLE

zs+lL

1/2+

3/2+

l/2-

3/2-

5/za+ See Table

IV at 40 MeV@

“P 4P

Partial

JP

MILLER

II.

Waves

(Column

2s 4D

(0.939, -0.048, 0.087,

?3 SD 4D

(-0.122, 0.481, 0.055, 0.153,

2P 4P

0.117, 0.013,

“P 4P 4P

0.68O)b 0.450 0.024 0.551y 0.433 -0.017 0.022

V at 46 MeV”

labels,

zs+lL, same as row labels.)

-0.036,

0.083

0.167, -0.014,

0.085 -0.002

0.291 0.005

0.337,

0.249

0.071, 0.027,

0.291 0.002

0.440,

0.243

0.377,

0.246

-0.043,0.078

NUCLEON-DEUTERON

ELASTIC TABLE

Partial

zs+lL

JP

1/2+

3/2+

l/2-

3/2-

5/2-

QSee

Table

zS+lL, same as row labels.)

labels,

(0.525, -0.008, 0.037,

0.323)* 0.214 0.007

0.024,

0.110

% 2D 4D

(0.036, 0.389, 0.060, 0.162,

0.315y 0.237 -0.005 0.020

0.161, -0.010,

0.087 0.000

2P 4P

0.130, 0.004,

0.213 0.009

0.310,

0.212

SP ‘P

0.071, 0.028,

0.213 0.003

0.345,

0.211

4P

0.319,

0.212

0.019,0.098

II.

Partial

zs+lL

Jp

1/2+

3/2+

1/2-

3/2-

5/2-

Table

at 77 MeVa

2s 4D

TABLE

“Gee

VI

Waves

(Column

227

SCATTERING

II.

Waves (Column

2s *D

(0.364, 0.024, 0.014,

0.23 1)” 0.137 0.003

% 2D *D

(0.060, 0.330, 0.060, 0.158,

0.251)” 0.174 -0.002 0.014

“P 4P

0.117, -0.002,

=P 4P 4P

VII at 95 MeV” labels,

zS+lL, same as row labels.)

0.041,

0.118

0.148, -0.006,

0.086 0.002

0.184 0.011

0.281,

0.193

0.055, 0.028,

0.184 0.003

0.293,

0.194

0.279,

0.194

0.036,0.102

228

REMLER

AND

TABLE Partial

JP

as+lL

1/2+

2s ‘D

3/2+

5 %D ‘D

l/2-

3/2-

s/2-

Osb See Table

Waves

(Column

(0.069, -0.069, -0.027,

labels,

*s+lL, same as row labels.)

0.051,

0.130

(0.026, 0.179)” 0.239, 0.084 0.053, -0.001 0.131, -0.010

0.098, 0.008,

0.087 0.006

2P 4P

0.052, -0.013,

0.138 0.015

0.187,

0.164

zP *P

-0.008, 0.024,

0.138 0.005

0.165,

0.159

‘P

0.171,

0.162

0.044,0.107

II. TABLE

as+lL

1/2+

3/2+

l/2-

3/2-

5/2a*b See Table

VIII at 146 MeVa

0.130)b 0.022 0.001

Partial

JP

MILLER

II.

Waves (Column

IX at 155 MeVa labels,

as+lL, same as row labels.)

?Y “D

(0.049, -0.019, -0.031,

0.127)” 0.029 0.003

0.050,

0.130

Yi eD *D

(0.036, 0.292, 0.052, 0.123,

0.174)b 0.073 -0.002 -0.013

0.088, 0.009,

0.087 0.007

aP 4P

0.043, -0.014,

0.135 0.016

0.175,

0.160

=P “P

-0.016, 0.023,

0.135 0.005

0.149,

0.155

“P

0.157,

0.158

0.041,

0.108

NUCLEON-DEUTERON

ELASTlC TABLE

Partial

JP

as+lL

1/2+

3/2+

l/2-

5/2a*b See Table

X at 198 MeV”

(Column labels, zs+lL, same as row labels.)

2s 4D

(-0.078, 0.077, -0.047,

0.127)L 0.186 0.009

0.041,

0.136

% *D *D

(-0.024, 0.124, 0.046, 0.102,

0.176)” 0.069 -0.006 -0.029

0.054, 0.019,

0.090 0.008

‘P

-0.006, -0.019,

0.128 0.020

0.109,

0.159

aP 4P

-0.060, 0.019,

0.128 0.006

0.077,

0.147

4P

0.089,

0.153

2P

3/2-

Waves

229

SCATTERING

0.027,

0.113

II.

The X + 1 points are on a reasonably smooth curve while the X + 1 + S points are somewhat erratic especially with respect to their real coordinate. There are a number of possible reasons for this ragged structure all of which undoubtedly contribute in some part. (i) Each X + 1 + S point is obtained by fitting to a different experiment and is thus subject to the vagaries of experimental error. (ii) The real part of the X + 1 + S amplitude is generally a small number obtained as the difference between two large numbers. (iii) It has been generally noted in phase shift analyses that the doublet phase is poorly determined by the cross section. Figure 14 is the corresponding Argand diagram for the quartet amplitude. We note that the X + 1 + S points fall to a much greater degree than previously on a smooth curve except for the 40 MeV point. Since it is clear that whatever are theoretical errors involved in this analysis, they will not fluctuate wildly between 36 and 46 MeV, we conclude that the 40 MeV data is in error. A similar phenomenon appears to occur near 155. It is also especially gratifying to note that in the case of this relatively well determined amplitude the low energy points appear to be smoothly connected to those at higher energies where the X + 1 + S theory gives by itself so much better a fit to the data. On both Argand diagrams we show points obtained by a previous phase shift analysis and by a solution of the Faddeev equations. The phase shift analysis was not expected to be accurate at these energies. The results of the (unpublished)

230

REMLER AND MILLER QUARTET + VAN OERS AND SROCKMAN X PEIPER

FIG. 14. Argand diagram for the quartet S-wave (see Fig. 13).

Faddeev calculation by Peiper [7] are in surprisingly close agreement with ours in the quartet case. In either case he differs from us by not much more than our points appear to fluctuate amongst themselves with respect to their lying on a smooth curve.

V. MEASURFXUENT OF THE DEUTERON D-STATE

PERCENTAGE

Intermediate energy elastic nucleon-deuteron scattering is an excellent probe of the high momentum components of the deuteron wavefunction because the deuteron needs just these components to stay intact after absorbing large momentum transfers. All realistic estimates of the deuteron wavefunction show that between internal momentum values of 0.3 to 0.7 Me the D-state momentum space deuteron wavefunction is larger than that of the S-state (see Fig. Al). Thus experimental data sensitive to such momenta should, with proper theoretical analysis, measure PD , the D-state percentage. In order to better understand what is involved here we have done the following. Using the X + 1 + S approximation we have fit the Postima and Wilson data with wavefunctions having 4 %, 6.963 %, and 12 % D-state percentages. These wavefunctions were manufactured from the Humberston-Wallace wavefunctions CJ+and v’D by changing them to ((1 - pD)/(l - .06963))1/2 yps

(58)

and for PD = .04, .06963, and .12 respectively. Thus for each value a separate fit to cross section and polarization was performed in D, Q parameter space.

NUCLEON-DEUTERON

ELASTIC

231

SCATTERING

Since the theory reproduces the data reasonably well for 8 5 150” we assume that these fits at these angles are relatively insensitive to the double scattering missing from the theory. The cross section fits are shwon in Fig. 15 over the angular range in which they differ from one another to any appreciable degree.

10-I , , , , , , , , , , , , , , , 30

60

so 120 C. M. ANGLE

150

I60

FIG. 15. Fits to the 146 MeV data using varying D-state percentage. The solid line is for 6.963 % D-state; the large dashed line is for 4 %; and the small dashed line is for 12 %. The data are from Refs. 30 and 31. The data of Ref. [30] have been multiplied by 0.92 (see [32]).

Curves of cross section versus PD at certain fixed angles were drawn by interpolating between these theoretical points. Data from the two 146 MeV experiments selected at angles within l/2” of the fixed angles were plotted at the PD values they each would imply. The results are shown in Fig. 16 and we note the following general features. (i) The measured cross sections from the two experiments agree within experimental error. (ii) Except for 150” and 155” the variation in PD values measured at different angles and experiments is consistent with experimental error bars. (iii) Disregarding the 150” and 155” points at which the X + 1 + S approximation clearly fails, there is definite clustering in the region 7 % 5 PD 2 9 %. (iv) The slopes of the theoretical curves are such that Au/a - ApD .

232

REMLER AND MILLER

0.60 F

0 A

0.07

-I 0

POSTMA + WILSON IGO ET. AL.

2

I 4 D-STATE

I 6 PERCENTAGE

I

I

6

IO

FIG. 16. Determination of the D-state percentage. The solid lines are cross section versus D-state percentage at fixed c.m. scattering angle the value of which labels the line. The data and histogram are drawn as described in the text.

To illustrate the clustering we have drawn a histogram at the bottom of Fig. 16, the height of which for each pD point is proportional to the number of experiments consistent with that pD value. A number of experimentally measured points at intermediate angles have been omitted for the sake of visual clarity. These do not affect our general basic results. These results lead us to two conclusions, one strong and one weak. Our weak conclusion is that pD lies between 7 % and 9 %. It is weak to the extent that the success of the theory in explaining the data is weak and also because we use some artificial wavefunctions. Note however that whatever error is incurred by using such wave functions it is not directly related to such relatively well known features such as the asymptotic S/D ratio or to any other of the well-known low momentum properties of the deuteron wavefunction. The relative momenta being probed here are of the order of .3 GeV. Our strong conclusion rests on the fact that it is hardly likely that the slopes of these curves will change in order of magnitude due to further theoretical refinements. Thus we claim to have demonstrated the feasibility

NUCLEON-DEUTERON

ELASTIC SCATTERING

233

of measuring PD by this method to a relative accuracy level of the same order as that of the experimental cross section points themselves provided that the double scattering integral is included in the theoretical analysis. To our surprise we found that the polarization was a good deal less sensitive than the cross section to variations in PO except possibly at the backward positive peak. Since our agreement with the data here is only qualitative we have not done the corresponding analysis. There are of course many other elastic scattering observables which have not been measured at these energies and it is quite possible that these taken in conjunction with the differential cross-section data will provide a much more sensitive measure of PD than any one of them alone. Finally we comment that the energy-angle window one has available to measure high momentum components is really quite limited. At energies below about 100 MeV, they are simply not being probed. At energies above about 300 MeV, the virtual production of isobars and other relativistic effects complicate the theoretical analysis to a degree not yet fully appreciated.

VI. SUMMARY OF RESULTS AND CONCLUSIONS 1. A careful calculation of the single and exchange nucleon-deuteron scattering amplitude has been performed. The results have been tabulated in a form which should be useful to future investigations. 2. The procedure for deriving contributions to the field theoretical multiple scattering series has been reviewed and illustrated. The procedure for relating these contributions to the deuteron wavefunction and the nucleon-nucleon elastic scattering data has been examined in a relativistic context. 3. It has been demonstrated that exchange, single and the isotropic part of all other terms in the multiple scattering series qualitatively account for all polarization and cross-section data between 30 and 198 MeV except for low energy polarization which can be fixed up by a very small addition of spin-orbit p wave. 4. It has been made at least plausible that the additional calculations of double scattering will change the aforementioned qualitative agreement to quantitative agreement throughout this energy range. 5. S-wave phase shifts obtained by fitting to experiment using this model have been obtained allowing comparisons to be drawn between different experiments. All experiments appear to lie on a smooth line except for one at 40 MeV. 6. The possibility of analysis of experiment [5] in this range as probes of the D-state wavefunction at high momenta has been examined and its utility in this regard been made plausible.

234

REMLER AND MILLER

7. Finally we emphasize the basic idea exploited in this work, that higher multiple scattering terms have lower partial wave content while lower multiple scattering terms have higher partial wave content, higher “structural content” and closer relationship to details of the nuclear wavefunction. Therefore, progress in investigating nuclear structure of the light nuclei using hadronic probes need not wait upon successful resolution of much more difficult complete multiple scattering problems.

APPENDIX

A: THE DEUTERON WAVE FUNCTION

The deuteron wavefunction is needed in the computation of both the nucleon exchange (Eq. (52)) and the single scattering (Eq. (53)) terms. The deuteron wavefunction is customarily decomposed into an S- and a D-state scalar wavefunction and spin spherical harmonics as given in Eq. (47) and Ref. [12]. We use the momentum space wavefunctions corresponding to the analytic wavefunctions of

0

(938

MEVK)

FIG. Al. The wavefunction of Humberston and Wallace [35], which was used in this work. Those are the scalar wavefunctions which multiply the appropriate spin spherical harmonics.

NUCLEON-DEIJTERON

235

ELASTIC SCATTERING

Humberston and Wallace [35]. Specifically, we use the wavefunction defined by the set of parameters for L = 4, N = 2 in Table 1 of [35]. The Humberston-Wallace wavefunction is an analytic approximation to the solution of the Hamada-Johnston potential [36]. The scalar wavefunctions & , +D are plotted in Fig. Al. Their argument, Q, is the magnitude of the three-momentum of either nucleon in the deuteron rest frame in units of the nucleon mass. APPENDIX

B: NUCLEON-NUCLEON

AMPLITUDES

The nucleon-nucleon amplitudes needed in the calculation of the single scattering integral equation (53) were obtained utilizing beta decay type invariant amplitudes M defined by TAP33P4

iP19P2)

=

z;c~IYPB

YP4

;Pl,P‘J"iYPl

,PJ

om%

,P4),

VW

where lis an isospin index and the sum is over the five current types: scalar, tensor, vector, axial vector, and pseudoscalar.

where

The neutron and proton amplitudes in terms of isospin amplitudes are (omitting the coulomb interaction)

I T I pn> = U/WI +P I T I P+ = = WWI

=
+ To), - To).

We proceed by first calculating each invariant amplitude for a grid of energyangle points using the phase shifts of MacGregor, Arndt, and Wright [37] (those shown and identified in Table VII of [37]). Then, when during the numerical integration of the single scattering integral, a value of T,(p, , pa ; p1 , pz) was

236

REMLER

AND

MILLER

needed, for any values of these four-momentum arguments, the currents jK were computed exactly, on-shell values of the invariant variables were fixed according to the prescription discussed in the text, values of the invariant amplitudes were obtained by interpolating between our grid values and finally T, was reconstituted as defined by Eq. (Bl). We spot checked these amplitudes using the available nucleon-nucleon cross section and polarization data from 50 MeV to 200 MeV lab kinetic energy and from 30” to 150“ c.m. scattering angle. The comparison is shown in Table BI. Note that we do not use pp amplitudes, so pp experiments are compared with our nn calculation. The differences observed are consistent with the omission of the coulomb interaction. Comparison of Experimental

TABLE BI and Calculated Nucleon-Nucleon

Lab energy”

c.m. angle

Reaction*

Observable

51.5 50. 51.5 98. 98. 98. 126. 126. 155. 174. 200. 199

31” 119” 149” 31” 31° 60” 33” 62” 89.8” 31.3” 31.5” 127.4”

oP w w PP np w nP np PP PP nP w

cross section polarization cross section polarization cross section cross section polarization polarization cross section polarization cross section polarization

“In MeV. b pp experiments are compared with c Cross sections are in mb/sr.

M

Observables

Experiment0 14.9 0.082 14.8 0.117 8.5 4.5 0.436 0.588 3.71 0.222 4.1 -0.125

* f f f f f f f & f A f

Calculationc

0.7 0.012 0.9 0.004 0.4 0.5 0.030 0.040 0.06 0.024 0.5 0.100

15.4 0.087 15.4 0.106 8.16 4.7 0.365 0.535 4.06 0.239 4.18 0.103

calculations

To conclude, we feel that the model of the nucleon-nucleon scattering amplitude used in the single scattering integral is as realistic as current knowledge permits. APPENDIX

C: EVALUATION OF THE SINGLE SCATTERING INTEGRAL AND TABULATED RESULTS

The single scattering amplitude was obtained by evaluating numerically the three-dimensional integral in Eq. (53). The origin of the variable of integration was taken to be the sticking factor point [38]. The three independent variables of inte-

NUCLEON-DEUTERON

ELASTIC

SCATTERING

237

gration were taken to be the radius and two spherical angles. Space inversion symmetry of the integrand was exploited to halve the number of integration points needed. Time inversion symmetry can also further halve this number but this was not done here. The azimuthal angle integration, which only had to be done over 180” because of the symmetry, was done with three equally spaced integration points with equal weights. The polar angle integration was actually done in the variable cos 6 using three-point Gauss-Legendre integration. This was empirically found to be satisfactory. The choice was motivated by the fact that doing the integral in cos 8 clusters integration points near the endpoints in 13.The integrand is large near the endpoints due to peaking there of either the initial or final deuteron wave function. The radial integral was done using an eight-point Gauss-Laguerre integration [39] which was constructed for integrals of the form m dx x2 exp( -ax2) f(x), s 0

where a is positive. We found that the value of the integral was insensitive to the value of the scale parameter a over a large enough range that we were able to integrate all combinations of the product of S- and D-state wavefunctions with one scale parameter, and hence one set of evaluations of the integrand. This was fortunate because the 4 and D-state wavefunctions (see Appendix A) have different rates of fall-off. If, as might have been expected, a different scale parameter and set of integrand evaluations were needed for the SS, SD, and DD wavefunction products, the calculation would have tripled in length. As it was, the 72 point integration scheme outlined above, using nucleon-nucleon amplitudes interpolated from a table as discussed in Appendix B, required about 20 minutes on an IBM 360/50. Such integrations were done every 30” from 0” to 180” for each energy of interest. Under these circumstances factors of 2 or 3 in computing time are significant. The number of integration points was set by varying the number for each dimension separately until the resulting value of the integral had converged to a few percent. Greater accuracy than that is not justified by other errors in our procedure. We observed that the lower energies required fewer integration points, and, in fact, the 31 MeV, 35 MeV, and the 0”, 30”, and 150” points at 46.3 MeV were done with four radial integration points instead of eight. A sample convergence test is presented in Table CI. We have selected 146 MeV, 60” as a representative laboratory energy and center of mass angle. At this energy and angle the differential cross section, nucleon polarization, and the deuteron tensor moments, Tl,, as calculated from the single scattering amplitude alone are tabulated for various combinations of numbers of integration points. We present the observables rather

238

REMLER

AND

MILLER

TABLE

CI

Convergence Test at 146 Mev, 60” Number of integration points Azimuthal Radial Polar 8 12 4 8 8

3 3 3 4 3

dojdsd

P

iTI

0.978 0.966 0.935 0.990 0.963

0.347 0.351 0.308 0.349 0.353

1.5%

1.8%

3 3 3 3 4

Variation excluding 4-point radial

Tall

TZl

T*,

0.222 0.225 0.202 0.227 0.226

-0.043 -0.043 -0.056 -0.040 -0.042

0.099 0.096 0.101 0.097 0.102

-0.223 -0.228 -0.226 -0.230 -0.224

2.4%

7.0 %

3.0%

3.1 %

0.6

0.7

--4 /' /

0.6

/ /

4

0.2

0.1

0.0 0

30

90

60 C.M.

120

150

160

ANGLE

FIG. Cl. Measure of distance off mass-shell, Ml.,,, - MB, as a function of laboratory kinetic energy and c.m. angle of the incident neutron, The solid lines are for 40 MeV, and the dashed lines are for 146 MeV. Bach line is marked with a radial integration point number. See Appendix C for further discussion.

NUCLEON-DEUTERON

ELASTIC

239

SCATTERING

than the scattering amplitudes themselves because the observables are more relevant, and it is easier to judge accuracy in terms of them. The deuteron tensor moments are defined and discussed in an accompanying paper. It was mentioned in the discussion surrounding Eq. (45) that the nucleon-nucleon amplitude must be extended off-shell in the process of calculating the single scattering integral. In order to lend some quantitative support to the approximation in Eq. (45) we present in Fig. Cl information about the range of ME2 and MF2 encountered in the numerical evaluation of T1 . The solid lines are for 40 MeV and the dashed lines are for 146 MeV. For each energy there are four curves, one for TABLE Single

CII

Scattering

Amplitude

0”

30”

60

90

120

150”

180”

+*, +1, +&, +1

-0.22850 -0.37955

-0.18782 -0.38817

-0.10441 -0.27832

-0.05425 -0.17041

-0.03123 -0.10292

-0.02205 -0.06475

-0.02054 -0.04307

+&, +1, +&, -1

-0.00049 0.00326

-0.00491 -0.00302

0.00047 -0.01307

0.00896 -0.01262

0.01385 -0.00800

0.01785 -0.00110

0.01699 0.01064

+$.

0.04909 -0.01289

0.04100 0.06554

0.00233 0.06876

-0.02147 0.04496

-0.02469 0.01824

-0.02205 0.00033

-0.04344 -0.08539

-0.03948 -0.05477

-0.04003 -0.03771

c, 4 a, b

+1,

-+,

0

-0.29303 -0.41761

0.06006 0.02460 -0.22903 -0.38048

-0.12121 -0.24922

To,&‘)

at 35 MeV

-0.06382 -0.14338

i-Q, 0, +*,

0

+s,

0, -+,

+1

0.13900 0.04599

0.11784 0.04098

0.08192 0.03026

0.05755 0.02192

0.04309 0.01720

0.03330 0.01284

0.02954 0.00780

++,

0, -4,

-1

0.04909 -0.01289

0.05664 0.02763

0.03246 0.06811

-0.00923 0.06691

-0.03096 0.03800

-0.02789 0.01140

-0.02205 0.00033

+*,

-1,

+a,

-1

-0.09732 -0.13818

-0.08452 -0.08327

-0.08962 -0.05964

-0.09350 -0.05363

+g,

-1,

-4,

0

-4,

+1,

-+,

$1

-0.35550 -0.46311

-0.27400 -0.38020

-0.15242 -0.23568

-0.09478 -0.13375

-0.08284 -0.08258

-4,

+1,

-4,

-1

-0.0@049 0.00326

-0.00110 -0.00477

0.00594 -0.01195

0.01447 -0.00940

0.01919 -0.00266

-0.29303 -0.41761

-0.21578 -0.29948

-0.11176 -0.15793

-0.06137 -0.07490

-0.04438 -0.04009

-0.04072 -0.03144

-0.04003 -0.03771

-0.22850 -0.37955

-0.16433 -0.23735

-0.08797 -0.11080

-0.04988 -0.05088

-0.03273 -0.03154

-0.02418 -0.03184

-0.02054 -0.04307

-g&o, -8,

-+, -1,

0 -+,

-1

-0.35550 -0.46311 0.13900 0.04599

-0.27594 -0.38720 0.11724 0.04009

-0.15528 -0.24369 0.08124 0.02807

0.05747 0.01612

0.04355 0.00790

0.03374 0.00512 -0.08879 -0.06036 0.02011 0.00453

0.02954 0.00780 -0.09350 -0.05363 0.01699 0.01064

240

REMLER

AND

TABLE

MILLER

C III

Single Scattering Amplitude T-(B) at c, d, a, b

+3,+1,+g,+1 +&,+1,+a,-1

146 MeV

0”

30”

60”

90”

120”

150”

-0.24135 -0.21699

-0.12027 -0.22684

-0.02384 -0.09497

-0.00733 -0.03314

-0.00276 -0.01252

-0.00236 -0.00360

-0.00362 0.00168

-0.00013 -0.00916

0.00657 -0.00644

0.00179 -0.00133

-0.00059 -0.00400

0.00428 -0.00733

0.01038 -0.00370

0.01877 0.01441

0.00693 0.01332

0.00093 0.00959

-0.00215 0.00483

-0.00218 -0.00051 0.00580 -0.00104

0.00211 0.00014 0.05488 -0.03944

0.04097 -0.00037

0

-0.21298 -0.21594

-0.07562 -0.15717

0.00320 -0.04770

0.00588 -0.01466

0.00419 -0.00793

0.00498 -0.00508

0, -4,

+1

0.01327 -0.04057

0.00034 -0.01405

0.00452 -0.00021

0.00859 0.00166

0.00714 0.00249

0.00273 0.00236

+g,

0, -4,

-1

0.05488 -0.03944

0.03912 0.00328

0.00954 0.01875

-0.00567 0.01249

-0.00862 0.00377

+&,

-1,

+g,

-1

-0.18572 -0.21673

-0.04371 -0.10505

0.02226 -0.01683

0.01370 0.00417

+;,

-1,

-*,

0

0.01327 -0.04057

-0.00125 -0.01438

0.00350 -0.00007

-4,

+1,

-g,

+1

-0.18572 -0.21673

-0.04132 -0.09099

0.02331 -0.00837

-+,

+1,

-4,

-1

-0.00355 -0.00284

-0.00683 0.00133

4

0, 40

-0.21298 -0.21594

-0.05449 -0.02624

0.00861 0.02451

0.00693 0.01673

--a,

-1,

-0.24135 -0.21699

-0.08086 0.03247

-0.01250 0.06422

-0.00515 0.03751

+$,

+1,

+a,

0, +*,

+a,

-:,o

-+.

-1

0.00211 0.00014

180”

-0.00082 -0.00190

-0.00563 -0.00116

-0.00218 -0.00051

0.00272 0.00686

-0.00321 0.00602

-0.00554 0.00364

0.00860 -0.00162

0.00753 -0.00459

0.00295 -0.00576

-0.00082 -0.00190

0.01446 0.00525

0.00346 0.00373

-0.00282 0.00219

-0.00554 0.00364

0.00536 -0.00753

0.01038 -0.00370

0.00498 0.00848

0.00565 0.00351

0.00580 -0.00104

-0.00265 0.01927

-0.00261 0.00803

-0.00362 0.00168

-0.00822 -0.00221

-0.00324 -0.00707

each of the four radial integration points used in this test case. The eight-point set of radial integration points extends about twice as far as the four-point set. Each curve shows the maximum of the quantity (M& - M2) as a function of the center-of-mass angle of the nucleon-deuteron scattering being calculated. (It should be clear that the range of MEa is exactly the same as that of MF2.) For any given radial integration point the values of are quite evenly spaced between the maximum plotted and zero for the integration over angles. As is expected the first radial integration point, which is nearest the sticking factor point, requires amplitudes very nearly on shell and each successive radial point requires amplitudes that are

NUCLEON-DEUTERON

ELASTIC

241

SCATTERING

further off shell. Also backward nucleon-deuteron angles require amplitudes that are more off-shell, but this trend is not as fast as might have been expected. To set the scale for interpretation of Fig. Cl, we note that a one pion plus one nucleon state has a mass squared of 1.32 (recall our unit is the nucleon mass) and the iV(1470) has a mass squared of 2.46. From Fig. Cl we see that M& certainly gets as large as the pion plus nucleon mass squared, especially for the eight-point radial integration. Concern about off-shell effects is therefore certainly justified. On the other hand, the great bulk of the integration points require smaller values of A&. The points near the sticking factor point, which contribute heavily TABLE Single

c, 4 a, b

Roaaa(6’) at 198 MeV

30”

60”

90”

-0.19494 -0.21018

-0.08095 -0.21033

-0.01052 -0.07392

-0.00524 -0.02177

0.00034 0.00149

0.00281 -0.01105

0.01010 -0.00370

0.00197 0.00062

0.05686 -0.05166

0.03214 -0.00814

0.01182 0.00197

0.00583 0.00409

0

-0.16322 -0.21519

-0.03460 -0.14071

0.01225 -0.03420 -0.00133 -0.00128

+1,

+-&, +1

+a.

+1,

+B,

-1

+a,

+1,

-g,

0

+s,

CIV

Amplitude

0”

+&,

+4,0,

Scattering

120

150”

180”

-0.00157 -0.00753

-0.00157 -0.00159

-0.00271 0.00060

-0.00069 -0.00226

0.00303 -0.00552

0.00843 -0.00363

0.00178 0.00382

-0.00138 0.00197

-0.00155 -0.00175

0.00487 -0.00973

0.00298 -0.00608

0.00466 -0.00479

0.00667 -0.00276

0.00446 0.00042

0.00429 0.00084

0.00063 0.00123

+*,

0, -8,

+1

0.01285 -0.04408

-0.00472 -0.01388

+t,

0, -&,

-1

0.05686 -0.05166

0.03088 -0.00481

0.00291 0.00654

-0.00510 0.00374

+i,

-1,

+*,

-0.13249 -0.22132

0.00229 -0.09340

0.03271 -0.01272

0.01400 0.00217

0.00357 0.00379

0.00019 0.00343

+$,

-1,

-to

0.01285 -0.04408

-0.00619 -0.01390

-0.00245 -0.00156

0.00400 -0.00214

0.00412 -0.00383

0.00048 -0.00522

-0.00289 -0.00213

-4,

+1,

-4,

+1

-0.13249 -0.22132

0.00359 -0.07858

0.03250 -0.00331

0.01403 0.00353

0.00387 0.00088

0.00034 -0.00054

-0.00081 0.00114

-4,

+1,

-Q.

-1

-0.00223 -0.00118

-0.00761 0.00720

-0.01027 0.00029

0.00298 -0.00727

0.00843 -0.00363

-0.16322 -0.21519

-0.01102 -0.01242

0.01949 0.02086

0.00746 0.00875

0.00475 0.00185

0.00587 -0.00069

0.00667 -0.00276

-0.19494 -0.21018

-0.03481 0.05173

0.00683 0.05886

0.00146 0.02818

0.00089 0.01253

-0.00084 0.00382

-0.00271 0.00060

--go, -4,

-*. -1,

-1

0 -4.

-1

0.00034 0.00149

-0.00573 -0.00045

-0.00544 -0.00604

-0.00406 -0.00238

-0.00289 -0.00213 -0.00155 -0.00175 -0.00081 0.00114

REMLER AND MILLER

242

to the integral, require values of M& q uite near 1. For these modest amounts off-shell any off-shell prescription should be satisfactory and our prescription seems especially simple and plausible. To conclude this appendix we present in Tables CII, CIII, and CIV tabulations of the calculated single scattering T-matrix, Tl , itself. Our main motivation in doing this is to provide future workers with a basis for detailed checks. Calculations of the single scattering amplitude involve a number of hard to verify conventions in addition to innumerable possibilities for computational error. We have found comparison with earlier works to be an invaluable aid in such problems. We point out immediately, however, that we estimate the uncertainty due to all numerical approximations to be about 10% for the large amplitudes and 100% for the smallest ones. We represent the single scattering T-matrix as T,,,,(e), where 8 is the scattering angle and a, 6, c, d are the spin components, in the pE X pp direction, of the entering nucleon, the entering deuteron, the final nucleon and the final deuteron, respectively. Parity invariance requires that all amplitudes for which c + d + a + b is odd be zero. Time reversal invariance gives Todab

= ei(c+d-a-b’*Tabcd(e).

These relations leave 12 independent complex amplitudes. We identify and tabulate the 12 amplitudes we have chosen in Tables CII, CIII, and CIV. They are tabulated as a function of center-of-mass scattering angle for laboratory kinetic energies of 36, 146, and 198 MeV. Because of space limitations the imaginary part is tabulated below the real part in each case. The normalization of these amplitudes can be inferred from Eqs. (lo), (1 l), and (12). They are in units of (nucleon mass)-2. APPENDIX

D: THE STICKING

FACTOR

APPROXIMATION

The sticking factor approximation mentioned earlier has been well motivated and discussed by Kowalski and Feldman [38]. In the context of the preceding appendix it may be thought of as a one-point evaluation of a factor in the integrand of the single scattering integral. Specifically, Eq. (53) is approximated by

where +L,L, are one of the scalar deuteron wavefunctions, I$~ or $D , plotted in Fig. Al. The remaining parts of the integrand in Eq. (53) are evaluated at (k) = MD,

+ &)/I

DE + DF I,

(W

NUCLEON-DEUTERON

and are denoted magnitude. The maximum of the At larger angles

ELASTIC

243

SCATTERING

by (U,,,,). In Eq. (D2) note that I DE + DF I is the four-vector sticking factor point defined in Eq. (D2) is the location of the integrand in Eq. (53) for small nucleon-deuteron scattering angles. this is no longer true. IO2

IO0

0

30

60

90

120

150

160

C. M. ANGLE

FIG. Dl. Comparison of sticking factor and integrated single scattering calculations at 40MeV. The solid line is the calculation of single scattering plus nucleon exchange of this work. That is, it is fully relativistic and the single scattering integral is evaluated numerically. The dashed line is our calculation of the sticking factor approximation to the single scattering integral, plus nucleon exchange. The boxes are the sticking factor approximation plus nucleon exchange of Kottler and Kowalski [4] as read from their published curve. The data are from Ref. [24].

In Fig. Dl we have plotted the data of [24], the sticking factor approximation of Kottler and Kowalski [4], our calculation of the sticking factor approximation, and our calculation of the single scattering integral. All the theoretical calculations include exchange scattering, but not the S-wave; that is, they are what we have been calling X+ 1. Kottler and Kowalski’s values are plotted as boxes to indicate the error involved in reading their published graph. There are differences between Kottler and

244

REMLER

AND

MlLLER

Kowalski’s calculation and our sticking factor calculation. For example wavefunctions and nucleon-nucleon amplitudes used are not the same, and formalism is basically nonrelativistic while ours is relativistic. However, differences should not be important at the energies of interest here, apparently they are not.

APPENDIX

the their the and

E: IMPROVED Low ENERGY FITS

The low energy group of fits to differential cross section using the X + 1 + S approximation described in the main body of this paper are only qualitative at large angles. That is, they are of correct shape and order of magnitude but are in relative error up to 50 %. Consequently, the polarization predictions, which are small, are completely wrong. The purpose of the side investigation reported in this appendix was to verify that the correction in the P-wave required to agree with experiment was of the order of magnitude expected from theoretical estimates and

C. M. ANGLE FIG. El. Fit to the c.m. differential cross section at 36 MeV using a phonomenological spin-orbit P-wave. The data are from Ref. [22]. The solid line is the fit from the main text. The dashed line is the result of a refit including the P-wave.

NUCLEON-DEUTERON

-0.4

ELASTIC

SCATTERING

245

1

E2. Fit to the neutron polarization at 36 MeV using a phenomenological spin-orbit P-wave. The data a.re from Ref. [23]. The solid line is the fit from the main text. The dashed line is the result of a refit including the P-wave. FIG.

therefore consistent with a double scattering contribution, and that the improved fit did not change the S-wave phase shift to any appreciable degree. It seemed likely that the best way to directly improve polarization would be by adding a term to TX+,+, which coupled nuclear spin to P-wave momenta. Thus we obtained Tx+l+s+p, given by Eq. (55). The cross section and polarization data at 36 MeV were simultaneously fit with this expression. The results are shown in Figs. El and E2 along with the data and the fits to cross section only described in the text. The evident dramatic improvement in polarization was effected by a change in PLs from zero (X + 1 + S approximation) to 0.023 + 0.033i. The numbers are in units of iW2 with T defined by Eqs. (9) and (10) of the text. In comparison the values of Q and D defined by Eqs. (3) and (4) of the text were Q = -0.158 + O.O42i, using X + 1 + S approximation Q = -0.158

D = 0.220 + 0.07Oi

and, + O.O41i,

using the X + 1 + S + P’ approximation.

D = 0.221 + 0.069i

246

REMLER AND MILLER

We note that the P-wave correction is uniformly considerably smaller than that of the S-wave and in fact corresponds in order of magnitude to that found by Sloan [9] in his model calculation for the P-wave content of double scattering. In this same reference it can be seen that a computation of TX + Tl + T2accounts for P and higher waves to high accuracy. Finally we note, as advertised, that the change in the fitted values of Q and D due to adding P-wave is negligible.

ACKNOWLEDGMENTS We are indebted to Dr. Pieper for sending us his unpublished results and allowing us to use them. We are indebted to Professor Igo for early information about his experimental results. One of us (E. A. Remler) is grateful for an informative discussion with Professor Sloan. Mr. John Wilson came up with the original suggestion for extrapolating the nucleon-nucleon T-matrix off-shell. He also provided us with an number of useful subroutines and data compilations. Special thanks go to the William and Mary Computer Center for extraordinary services rendered.

REFERENCES 1. R. AARON, R. D. AMADO, AND Y. Y. YAM, Phys. Rev. 140 (1965) B1291. 2. I. H. SLOAN, Nucl. Phys. Al68 (1971), 211. 3. N. M. QUEEN, Nucl. Phys. 55 (1964), 177. 4. H. KO~LER AND K. L. KOWALSKI, Phys. Rev. 138 (1965), B619. 5. E. A. REMLER, Ann. ofPhys. 67 (1971), 114. 6. E. A. REMLER, Phys. Rev. 176 (1968), 2108. 7. S. C. PIEPER, Perturbative Calculation of Spin Observables in Nucleon Deuteron Elastic Scattering, Case Western Reserve University preprint, January 1972. 8. I. H. SLOAN, A Three-Nucleon Scattering Calculation Using Perturbation Theory, University of Maryland preprint, 1972. 9. I. H. SLOAN, Phys. Rev. 185 (1969), 1361. 10. E. A. REMLER, R. A. MILLER, AND J. W. WILSON, Bull. Amer. Phys. Sot. 17 (1972), 438. 11. E. A. REMLER, Investigation of the Nucleon-Deuteron System, in “Proceedings of the Third International Conference on High Energy Physics and Nuclear Structure, New York,” 1969, p. 324 Plenum Press, New York, 1970. 12. E. A. REMLER, Nucl. Phys. B42 (1972), 56. 13. E. A. R!&&ER, Nucl. Phys. B42 (1972), 69. 14. M. L. GOLDBERGER AND K. M. WATSON, “Collision Theory,” Wiley, New York, 1964. 15. S. MANDELSTAM, Proc. Roy. Sot. London, Ser. A 233 (1955), 248. 16. A. KLEIN AND C. ZEMACH, Phys. Rev. 108(1957), 126. 17. F. L. GROIN, Phys. Rev. 140 (1965), B410. 18. W. T. H. VAN OERS AND K. W. BROCKMAN, JR., N&Z. Phys. A92 (1967), 561. 19. J. C. ALDER, W. D~LLHOPF, C. LIJNKE, C. F. PERDRISAT, W. K. ROBERTS, P. KITCHING, G. Moss, W. C. OLSEN, AND J. R. PRIEST, submitted to Phys. Rev. 20. C. C. KIM, S. M. BUNCH, D. W. DEVINS, AND H. H. FORSTER,Nucl. Phys. 58 (1964), 32.

NUCLEON-DEUTERON

ELASTlC SCATTERING

247

21. V. J. ASHBY, UCRL-2091 (1953). 22. J. L. ROMERO, J. A. JUNGERMAN, F. P. BRADY, W. J. KNOX, AND Y. ISHIZAKI, Phys. Rev. C2 (1970), 2134. 23. S. N. BUNKER, J. M. CAMERON, R. F. CARLSON, J. R. RICHARDSON, P. ToMAs, W. T. H. VAN OERS, AND J. W. VERBA, Nucl. Phys. All3 (1968), 461. 24. J. H. WILLIAM?, AND M. K. BRUSSEL, Phys. Rev. 110 (1958), 136. 25. H. E. CONZETT, H. S. GOLDBERG, E. SHIELD, R. J. SLOBODRIAN, AND S. YAMBE, Phys. Lett. 11 (1964), 68. 26. A. R. JOHNSTON, W. R. GIBSON, E. A. MCCLATCHIE, J. H. P. C. MEGAW, AND R. J. GRIFFITH& Phys. Lett. 21 (1966), 309. 27. M. DAVISON, H. W. K. KOPKINS, L. LYONS, AND D. F. SHAW, Nucl. Phys. 45 (1963), 423. 28. 0. CHAMBERLAIN AND M. 0. STERN, Phys. Rev. 94 (1954), 666. 29. M. POULET, A. MICHAU~WICZ, Y. LEGUEN, K. KURODA, D. CRONENBERGER, AND G. COIGNET,

J. Physiq. 26 (1965), 399. 30. H. POSTMA AND R. WILSON, Phys. Rev. 121(1961), 1229. 31. G. IGO, J. C. FONG, S. L. VERBECK, M. GOITEIN, D. L. HENDRIE, J. C. CARROLL, B. MCDONALD A. STETZ, AND M. C. MAKINO, UCLA preprint, summer 1972. 32. J. N. PALMIERI, Nucl. Phys. A188 (1972), 72. 33. K. KURODA, A. MICHAL~WICZ, AND M. POULET, Nucl. Phys. 88 (1966), 33. 34. R. E. ADELBERGER AND C. N. BROWN, Phys. Rev. D5 (1972), 2139. 35. J. W. HUMBER~TON AND J. B. G. WALLACE, Nucl. Phys. A141 (1970), 362. 36. T. HAMADA AND I. D. JOHNSTON, Nucl. Phys. 34 (1962), 382. 37. M. H. MACGREGOR, R. A. ARNDT, AND R. M. WRIGHT, Phys. Rev. 169 (1968), 1128. 38. K. L. KOWALSKI AND D. FELDMAN, Phys. Rev. 130 (1963), 276; and Appendix D. 39. P. RABINOWITZ AND G. WEISS, Math. Tables Aids Comp. 68 (1959), 285. 40. J. D. SEAGRAVE, The Elastic Channel in Nucleon-Deuteron Scattering, in “Proceedings of the

International Birmingham,

Conference on the Three-Body Problem in Nuclear and Particle Physics, England, 1969,” p. 41, North Holland, Amsterdam, 1970.