Nature of the amplitudes missing from adiabatic distorted-wave models of medium energy (d, p) and...

Nuclear Physics AS05 (1989) 26-66 North-Holland. Amsterdam

NATURE OF THE AMPLITUDES MISSING FROM ADIABATIC

DISTORTED-WAVE MODELS OF MEDIUM ENERGY (d,p) AND (p, d) REACTIONS*

R.C. JOHNSON, E.J. STEPHENSON’ and J.A. TOSTEVIN

Department of Physics, Unioersity of Surrey, Guildford, Surrey GU2.5XH, UK ’ l~diana UniniDersity Cyclotron Facility, Bloomington, IN 47405, USA

Received 25 April 1989

(Revised 15 August 1989)

Abstract: Simple features in the reaction mechnism of (d, p) transfer reactions at medium energies (near 100 MeV) are clarified and used to illustrate problems with an adiabatic distorted-wave model of

the process, currently the most reliable calculation available. These problems are most apparent

for j, = I, -4 transfer, for which additional simplicities are obtained. In an effort to understand

the nature of these problems, we will demonstrate that semi-classical approximations not usually

associated with light-ion physics are a quantitatively useful tool in this energy regime. The

combination of these theoretical ideas with the availability of measurements of polarization

observables over a wide angular range provides new and unique insights into the nature of the

shortcomings of the adiabatic model. In particular, large amplitudes arising from specific combina- tions of orbital and spin angular momentum quantum numbers can be identified as missing from

these calculations. The prospect that such amplitudes could be produced by an improved treatment

of deuteron breakup is discussed briefly.

1. Introduction

In recent years, the theoretical treatment of the breakup continuum in deuteron-

induced scattering and reactions has received increasing attention. For two recent

and extensive review articles see ref. ‘). The discretized continuum coupled channels

(CDCC) techniques, pioneered by Rawitscher 2, and the Pittsburgh group 3), have

been refined by the Kyushu collaboration ls4) to the point where routine calculations

are feasible, if computationally expensive. Perhaps as important, the CDCC method

has provided benchmark calculations by which more ~omputationally efficient but

approximate breakup treatments, such as the adiabatic approximation 5), can be

assessed ‘*6).

Whiie for elastic scattering and breakup studies calculations have advanced to a

very sophisticated level, including D-state and full spin-orbit distortion effects, the

CDCC has yet to be applied to transfer reactions in anything but the zero-range

approximation ‘). Thus, only transfer for np relative S-waves is included. It is well

known however *), particularly at higher incident deuteron energies 9), that finite-

range and deuteron D-state effects are vital to reproducing the observed spin

l This paper is dedicated to the memory of Lionel J.B. Goldfarb, who was both a colleague and friend, and who was to one of us a valued teacher.

0375.9474/89/SO3.50 @ Elsevier Science Publishers B.V.

(Noah-Holland Physics Publishing Division)

R.C. Joimson et aL / Adiabatic distorted-~vave approximation 27

dependence of (d, p) reactions. The simpler adiabatic calculations do allow these

finite-range effects to be included in addition to the spin-dependent distortions;

however the method treats only relative S-wave breakup and this approximately lo).

In comparisons with intermediate energy measurements [as in ref. “) for example],

the omission of D-wave breakup appears to be less serious for many transitions

than the omission of spin-dependent distortions or stripping from the deuteron

D-state. Thus the adiabatic approximation becomes the best currently available

theoretical calculation, and the one whose reliability we will explore in detail.

In general, for reactions on medium to heavy mass targets, both CDCC and

adiabatic calculations provide a reasonable quantitative description of the

measurements 1*6). It is usually not known, however, whether the discrepancies that

invariably remain arise from uncertainties in our detailed knowledge of the optical-

model and other parameters that enter the calculation, or whether there is some

more basic shortcoming of the theory, for example the omission of some definite

physical process.

In this paper we will investigate this question in detail for bombarding energies

near 100 MeV. Here the dynamics of the reaction lead to a number of simplifications

that will be exploited to probe the details of both the reaction and our theoretical

understanding of it. We will show that deficiencies in the transfer reaction amplitudes

obtained using the adiabatic approximation point toward a process missing or poorly

treated in these calculations; using information derived from the experimental

measurements, we will characterize this missing process. It is our hope that this

characterization will lead not only to an improved theoretical treatment and under-

standing of the reaction but also indicate the nature of the shortcomings of the

computationally efficient adiabatic method, leading ultimately to improved agree-

ment at all energies. In a wider context, these characteristics must be met by any

model that hopes to provide a detailed understanding of transfer reactions.

Indications of difficulties with medium energy transfer reaction calculations have

existed for some time. Angular distributions, especially those of vector analyzing

powers in (p, d) reactions, show poor qualitative agreement with the data 12).

Spectroscopic factors too often show variations of as much as an order of magnitude

with changing bombarding energy 13). In this situation, it is often difficult to extract

neutron 2, andj, values, even for strong transitions, and little credible spectroscopic

information can be obtained 13). One scheme for dealing with these difficulties is to

constrain the r.m.s. radius as well as the binding energy of the neutron bound-state

wavefunction which enters the adiabatic calculation. While this modification does

help to lessen the energy dependence of the deduced spectroscopic factors, it has

little effect on the quality of the angular distributions 13). Such effects as inelastic

channel coupling 14) and antisymmetrization “) (or medium modifications) intro-

duce only minor corrections at these energies.

Earlier work has suggested that these problems might be connected with the poor

angular momentum matching associated with low-l,, transitions ‘6) [see also

28 R.C. Jahnson et af. / Adja~atie disiorted-cave ~~~roxjrnat~o~

refs. 123’3)] at medium energies. A recent study of the highly mismatched

“%n(d, p)“‘Sn (g.s., I,, - 0) reaction ‘), however, did not indicate any inadequacy

in the adiabatic description. Unexpectedly, those transitions ‘r7r4) with large &,,

where the matching is good, were the ones which indicated serious problems with

the theoretical calculations. This was especially true when the transferred neutron

spin coupling was j, = I, - 4. These calculations were performed in finite range. They

included the deuteron D-state and S-wave breakup, through the adiabatic

approximation ‘,‘O), and utilized standard non-locality corrections I’).

It is important that this large 1, study included measurements of a number of

polarization observables over a wide angle range, including states with j, = 1, -4.

Using semi-classical ideas 18), the reaction mechanism may be divided into two

pieces, one associated with flux passing on the near side of the nucleus (toward the

detector) and another on the far side. It is possible to show that for such transitions

the dynamics favors the far-side components and simple spin couplings, and very

few of the reaction amplitudes should be large, leading to a number of relationships

among these normally independent polarization observables. Most of these sim-

plifications occur forj, = I, - f transitions; since these transitions also give the greatest

difficulties theoretically, we will emphasize them in our discussion of the spin-

dependent results. Although not satisfied exactly, these relationships are obeyed to

good precision by the quantum mechanical calculations. The fact that these relation-

ships are not satisfied by the data thus casts doubt upon the assumptions of the

semi-classical arguments, namely coplanarity of the bound neutron orbit with the

asymptotic reaction plane and far-side dominance in the amplitudes, as the source

of the disagreement. The presence of an interference pattern in the measured

analyzing powers, a ~hara~teristi~ of interference between near- and far-side ampli-

tudes, indicates that, at the very least, the assumption of far-side dominance is not

valid.

Semi-classical ideas have been used successfully in the study of heavy-ion reac-

tions, but their use for light projectiles is not commonplace. We will begin with a

justification of the semi-classical approach, showing that the combination of large-l,

transfer with higher bombarding energies produces theoretical reaction amplitudes

that match closeiy the character expected from the semi-classical assumptions.

However, in order to identify any new amplitude (arising for instance from a

neglected reaction channel) which may be needed, we must extract additional

information from the measurements. We will use the semi-classical approach to

model the experiment in such a way that the amplitudes generated by existing

theoretical calculations are easily identified. The remaining components of the

empirical amplitude correspond to new processes, not part of the current theory,

whose properties we can characterize through the semi-classical relationships.

It is important to stress at the beginning that we do not intend to use a semi-classical

model as a substitute for full quantum mechanical calculations. Rather, we use the

model, containing many fewer parameters than a conventional theoretical calculation

R.C. Johnson et al. / Adiabatic distorted-wave approximation 29

(optical-potential parameters etc.), and deduce values for the parameters from the

angular distributions of measured observables. The role of the semi-classical

approach is to interpret these parameters. It establishes a connection between the

parameters and transition amplitudes with specific quantum numbers. The particular

values of parameters introduced into the model will depend on the incident energy

and the particular transition considered. As such, the semi-classical mode1 is an

important diagnostic tool and a source of insight into the full quantum mechanical

adiabatic calculations.

We will thus show that there are amplitudes, currently missing from the calcula-

tions, which are comparable in size to those already present theoretically. Such

amplitudes, whether from a neglected process or an inaccurate treatment of included

channels, represent a serious failure of the adiabatic model. Coupled-channel models

of the breakup continuum will be examined briefly to assess the prospect that this

improved treatment of the continuum contains the missing amplitude.

2. Semiclassical character of medium energy transfer reactions without spin

The scattering of medium energy protons and deuterons from massive targets is

characterized by large spin-orbit effects ‘9320). These distortions are eivdent in the

analyzing powers for (d, p) and (p, d) transfer reactions “) as well. We will show,

however, that they do not change the essential features of the problem we are

studying. Rather, we can use them to separate the differential cross section into

partial cross sections for individual spin-projection channels where the distorting

forces have substantially different strengths. This is a critical feature of our analysis.

In order to develop the semi-classical approach and to understand how the concepts

of coplanarity and far-side dominance appear theoretically, we will begin with an

examination of the unpolarized cross section using calculations without spin-depen-

dent distortions. Later we will return to develop fully the consequences of these

features for the analyzing power and polarization.

At lower bombarding energies, the I, transfer essentially determines the scattering

angle at which the cross section is largest, making I, assignments to particular states

rather unambiguous. At such energies entrance and exit channel flux is present

around the entire nucleus, and interference between asymptotic amplitudes arising

from the two sides of the nucleus (relative to the detector) show up as strong

oscillations in the cross section and analyzing power angular distributions. As an

example from intermediate energies, cross section angular distributions for the

“‘Sn(d, p)“‘Sn reaction at 79 MeV bombarding energy are shown in fig. 1 for

transitions with I, transfer values ranging from 0 to 5. The interference pattern

present for the I, = 0 transition gradually disappears with increasing I, value. The

transitions with I, = 4 and 5 are well matched in angular momentum at the nuclear

surface, and present angular distributions that fall exponentially with angle. Little

I, dependence is present, and I, assignments cannot be made on the basis of such

R.C. Johnson et al. / Adiabatic distorted-wave approximation

‘i6Sn(d,p) “‘Sn IO’ ,,

Ed =79 MeV

I I I I

'"-50e 8 cm.

Fig. 1. Cross section angular distributions for the “‘Sn(d, p)“‘Sn reaction to the four lowest energy states in “‘Sn. The I, and j, transfer are given for each angular distribution, along with the spectroscopic

factor obtained from the distorted wave calculations described in the text. Those calculations are shown

for each case as a solid line. The cross section angular distributions calculated separately from the

far-side (dashed) and near-side (dotted) amplitudes are included.

data. The improved matching is evident in the rise in the magnitude of the cross

section with 1, (note that the spectroscopic factor, included for all transitions in

fig. 1, is about 0.1 for the g7,2 transition). Because of the attractive mean field forces

experienced by the deuteron and proton at the nuclear surface, this rise is manifest

primarily in the far-side amplitude. A decomposition of the calculated cross section

(full curves) into its near-side (dotted) and far-side (dashed) pieces, using the

method of Fuller Ix) (to be discussed in more detail later) is also shown in fig. 1.

The variation of the near- and far-side components, at O,.,. = 40”, as a function of

1, transfer is shown in fig. 2, assuming a spectroscopic factor of unity in each case.

R.C. Johnson et al. / A~iabufic d~storfed-wffve ap~ro~~mafion 31

Fig. 2. The “?3n(d, p)“‘Sn reaction cross section calculated at 0,,,, = 4O’and with a spectroscopic factor

of unity for the transitions shown in fig. 1. The solid points and lines show values based only on the

far-side amplitudes and the open points and dashed lines show values based only on the near-side

amplitudes.

It is clear that, as the far-side amplitude rises and the near-side contribution remains more stationary, the interference pattern will disappear. There is no indication that the near- and far-side contributions themselves display any interference oscillations.

These observations can be understood within the framework of standard theory and modefled using semi-classical ideas. In the absence of spin-orbit forces in the proton and deuteron channels and in the zero-range (or local energy) approximation 2’), the unpolarized differential cross section is an incoherent sum of partial cross sections corresponding to definite values of the projection quantum number, A,, of the transferred neutron orbital angular momentum, 1,. Because we wish to examine the concept of coplanarity in detail, it will be pa~icularly convenient to choose the quantization axis for A, to be along the direction normal to the scattering plane, i.e. along k,x k, [the Madison convention *‘) y-axisI_

The amplitude corresponding to each partial cross section, B,Jk,, k& can be separated into two pieces that may be interpreted in the semi-classical limit as the amplitudes arising from the near ( - ) and far ( + ) (away from the detector) sides of the nucleus

B /,A, = B!,ICh), + &,;I, . (1)

This separation, as elaborated by Fuller 18) and Dean and Rowley 23), is made by replacing the associated Legendre function appearing in the partial-wave sum with

32 R.C. Johnson et al. / Adiabatic distorted-wave approximation

either P$L’ or Pj,’ where

P,,(cos 6) = Pi;‘(cos e) + Pj,‘(cos e) )

P{~‘(cos 0) =;[P,,(cos tl)T if Qlm(cos e)] . (2)

The sums containing P$i’ are then associated with the B(*), respectively. The near-

and far-side interpretation works best for large partial waves and at scattering angles

sufficiently far from 0” (sin 8 B l/Z). Expressions for the amplitudes B$,i,‘, in terms

of the transfer reaction radial integrals have been given in refs. “,24) and are derived

most straightforwardly using the techniques of ref. 23). The result is

(3)

where f= (21+ 1)“2 and the radial integrals RI,,,,, depend on the deuteron (1,) and

proton (12) orbital quantum numbers as well as the transferred neutron angular

momentum (I,). Note that the arguments of the neutron angular wavefunction Y,.A\,

correspond to points along the x-axis on the near side, (8,, 4,) = ($r, &r), and

far-side, (0”) 4,) = ($r, --$I-) of the nucleus. In the zero-range limit the radial

integrals can be expressed in terms of the deuteron and proton radial wavefunctions,

u,,(r) and u,,(r), and the bound-state neutron form factor x,,.(r) as

with A the target mass number. Various constants, such as the zero-range normaliz-

ation factor and the spectroscopic factor, have been absorbed for this discussion

into A,,, . The functions Yi,:’ (8,O) are given by

yj;, = 1

’ %rr(sin e)1/2 exp[*i((l+;)&$r)]

in the limit of large angular momentum 1.

Eq. (3) shows that the neutron transfer with projection A,, along the normal to

the asymptotic reaction plane is associated with radial integrals in which the

difference between deuteron (I,) and proton ( 12) orbital angular momenta is + A,,

for reactions that occur on the far-side of the nucleus. This is as one would expect

in a classical orbit picture of a (d, p) reaction in which the deuteron, proton and

neutron orbits are coplanar and rotating about the nucleus in the same sense under

the influence of attractive forces. Likewise a difference of -A, is obtained for

near-side reactions. Again in the simplest picture, the deuteron, proton and neutron

orbit about the nucleus in the same sense, but in a direction opposite to that for

R.C. Johnson et cd. / Adiabatic distorted-wave op~roxi~otioo 33

far-side reactions. The differential cross section is

and the property of incoherence implies that there can be no interference between

near-side and far-side amplitudes corresponding to different values of A,. For a

single value of An any interfering near-side and far-side components arise from

neutron waves travelling in the same sense about the nucleus. This point will emerge

later when we need to discuss the neutron-proton relative momenta involved in the

missing amplitude. When such an interference appears for a particular value of h,,

it must thus arise from the near-side and far-side projections of radial integrals [see

eq. (311 with different values of 12--2).

At medium energies, the dynamics associated with angular momentum matching

will strongly favor the ~ont~bution of only the extreme values of h, = *Z,, in the

cross-section sum of eq. (6). This reflects the surface peaked nature of the stripping

reaction where the partial waves of importance approximately satisfy 1, = kdr and

l2 = k,r where kd and k, are the deuteron and proton asymptotic momenta and Y is

the radius to a point on the nuclear surface. The largest radial integrals will satisfy

1, - 1, = ( kd - k,)r, which, since k, = v?!k,, assumes a value near 5 for the data of

fig. 1. When the angular momentum of the transferred neutron is near or equal to

this value, we would expect the dominant terms in the sum of eq. (6) to be the

far-side projection I.$_,” = B’,“Tb and the near-side projection BI,_,” = I$?,” of the . .

dominant radial mtegrals Rr,i,jr,+r,i. Thus near-side/far-side interference is strongly

suppressed in well-matched medium-energy (d, p) transitions. An inspection of the

cross sections in fig. 1 supports this trend.

Conversely, for an I, = 0 transition, both the near-side and far-side components

of IQ0 involve the same (badly-matched) radial integral RI,DI,, and strong interference

effects are expected at all scattering angles (as observed in fig. 1). The period of

oscillation is governed by the value of f2 for the largest radial integral according to

n/l2 [ref. ““)I. The theoretical peak at i2 = 13 [fig. 10 of ref. “)I gives a period of 14”,

corresponding closely to the spacing of the interference minima in fig. 1.

So far, we have made use only of the fact that the radial integrals peak at large

values of 1, and l2 corresponding to neutron transfer at the nuclear surface. This

property results in the dominance of the radial integrals R,2,nCi2tr”), and removes

from the cross section all contributions other than the coplanar ones where h, = f &, . Later we will examine the I, = 4 transition in detail as the spin coupling for the

transferred neutron proves advantageous to our investigation in that case. Fig. 3

shows the five non-vanishing components of the cross section IB,,h,lZ, as a function

of angle, calculated in the zero-range approximation (no deuteron D-state) and in

the absence of deuteron and proton spin-orbit coupling. Only these five satisfy the

parity conservation condition (-l)An = 1 [ref. ‘“)I. The optical potential parameters

of the adiabatic calculation are given in table 1. The deuteron parameters were fitted


e c.m

Fig. 3. Angular distributions of the cross section components /&,}” for all parity-allowed values of A,

between 4 and -4. The amplitudes for the angular distributions were taken from a zero-range distorted

wave calculation without deuteron and proton spin-orbit coupling.

to the adiabatic potential derived from the folding prescription lo) based on the

nucleon scattering potentials of ref. 28), and the outgoing proton potential was chosen

to reproduce proton elastic scattering measurements from “‘Sn at 83 MeV [ref. “)I.

The two largest amplitudes at nearly all angles correspond to h, = -f4, supporting

the contention that in the theoretical picture, the reaction is coplanar.

To understand far-side dominance (or why the A, = 4 contribution is much larger

than the A, = -4 term), we need more detailed information on the way in which

the radial integrals, and in particular their phase, varies for those important values

of 1, near the nuclear surface. The largest contributions to the sum for Bi,“x), in eq.

(3) are obtained when the product Y (*)R has a stationary phase point as a function

of &. For the far-side case, the phase of Y(+) in eq. (5) increases with 12, and the

sum will be large only if the phase of R shows a corresponding decrease with 12.

Fig. 4 shows an Argand plot of those radial integrals with I, - Z2 = 4. A large looping

behavior is clearly evident with a steady decrease of the phase of R with increasing

12. In the same way, the near-side piece is suppressed since the phase of Y(-‘, like

R, decreases with I*, and a large cancellation is present in that sum. These effects

are evident in the enhancement of the A, = 4 contribution shown in fig. 3 and the

corresponding suppression of the h, = -4 term.

R.C. Johnson et al. / Adiabatie distorted-wave approximation

TABLE 1

Optical-model parameters

35

deuteron proton

(adiabatic) (elastic)

=) “) 2X) “)

neutron

(bound)

real central (Woods-Saxon)

V 88.836 38.28 r 1.1636 1.177 1.25

a 0.7969 0.782 0.65

imaginary central (WS and deriv. WS)

w, 13.349 8.45

WD 4.071 0.67

rw 1.2926 1.322

ow 0.6685 0.637

real spin-orbit (Thomas) V S.0. 6.2 6.366 6.0

r, 0 1.01 1.003

a 5” 0.75 0.948

Coulomb radius

rc 1.25 1.25

“) The potential form factors follow these references.

“‘Sn (d, P)“%n (7/Z+) I I

IP

Fig. 4. Argand diagram showing the real (horizontal axis) and imaginary (vertical axis) parts of the

radial integrals where I, -% = 4 and 2. The values noted give the proton (1J partial wave number. The

radial integrals were taken from a zero-range distorted-wave calculation without deuteron and proton

spin-orbit coupling. The curves are a guide to the eye.


The rapid decrease in the size of the radial integrals as one moves away from the

well-matched 2, - Iz = 4 case is also illustrated in fig. 4 where the pattern for I, - 1; = 2

is shown. Other values of this difference yield radial integrals smaller still by at

least a factor of two in magnitude from the peak of the I, -12=2 term. Thus, the

trends expected for a coplanar reaction, seen in the magnitudes of the scattering

amplitudes in fig. 3, can be understood in terms of the phase behavior of the

underlying radial integrals.

3. Semiclassical model of reaction amplitudes without spin

We ultimately wish to address the question of the interference pattern in the

angular distributions of the (d, p) analyzing powers shown in ref. ‘I). The arguments

of the previous section make it clear that the theory presents us with a coplanar,

far-side-dominant picture. Unless there is also a near-side amplitude of magnitude

comparable to the far-side component, no interference pattern will be present. That

near-side component is apparently absent from our theory, as our later presentation

will show. Thus we must look not to theory but to the experimental measurements

themselves to ascertain the magnitude and character of the missing near-side ampli-

tude. To learn anything about this near-side component from the measurements,

we will need a way to represent it with a model containing near-side pieces that

can be adjusted independently of the far-side term. In principle, the theoretical

adiabatic calculations might be altered to make this so, but a means of achieving

this through the reaction calculation input, consisting of disto~ing potentials, bound

state wavefunctions, and a neutron-proton interaction, is not readily apparent.

Instead, the smooth and simple behavior of the I, - l2 = 4 radial integral in fig. 4

suggests that it may be modelled by a simple pole structure in the complex Z-plane.

An adjustable complex amplitude can be made to match the size of the radial

integrals, while the I-value of the maximal radial integral determines the real Z-axis

coordinate of the pole. The imaginary coordinate and the order of the pole must

then be chosen so as to reproduce the l-dependence of the magnitude and phase

of the dominant radial integrals. Naturally the location of the pole in either the

lower or upper half-plane determines the sign of the phase advance with 1, and

hence whether it contributes to near-side or far-side scattering. Thus, this simple

pole model description has the potential to represent each of the features of the

ampiitudes discussed in the previous section. Note, however, that the pole structure

of this model may have no simple interpretation in terms of singularities of the

exact S-matrix of the reaction.

Such a model is useful only insofar as it provides a quantitatively accurate picture

of either the theory or the data. In this section, we will present two analytic forms,

one with a single pole and another with an infinite set of poles, in order to illustrate

that there exists a close connection between the model parameters and features of

the radial integrals and reaction cross section. Finally, by matching the multiple

R.C. Jo~nsan et al. / Adiabatic d~stvrted-have approxi~afio~ 37

pole form to the calculated I, -Z, = 4 radial integrals of fig. 4, we determined how

well this model can in fact describe the full theoretical cross section.

The radial integrals corresponding to a single pole of any order n located at

L, + $iT may be written as a function of 1, in the form

R bl”(b+l”) - - Ro (7)

where R. is a complex amplitude and L2 and r are real. The parameter L2 clearly

determines the value of I2 at which R reaches its maximum, while r, together with

n, determines both the rate of phase advance with l2 and width of the dist~bution.

The relationship between phase advance and width can be altered by appropriate

choice of the order of the pole, n. With the radial integrals so parametrized, the

scattering amplitude of eq. (3) may be obtained in closed form by the use of

semi-classical approximations; the &-sum is converted to an integral along the entire

real 12-axis in which the large-l approximation to Y(*) in eq. (5) is made. If r > 0

the pole is located in the upper half-plane and the amplitudes are proportional to

Bl I =Bj:)a @7-i

“” ” ” (sin @)“2 exp [i(Lz+i)B] exp [ -@3],

B, -, = l3-2, =o nn nn 7 (9)

and the radial integral produces only a far-side amplitude. Conversely, if r < 0, the

pole is in the lower half-plane

B ‘“1” = Bjn;;=O , (10)

B, __I = l3-2, cc On-’ On nn (sin 1))“2 exp [-i(L2+5)B] exp [-:-p-p], (11)

and only a near-side amplitude results. In either case, the main feature of the cross

section will be an exponential decrease as exp (-/r/e). The behavior at small

scattering angles will be modified by the additional dependence upon sin @ and the

choice of the order of the pole, n. If there is more than one pole term in the model

with different values of L2 and r, an interference pattern will appear in the cross

section. Given two poles with real axis positions L2 and Li and both poles in either

the upper or lower half-plane, the oscillation pattern will be sin ( LZ - LJ)B and a

slow &dependence will result. If, on the other hand, the poles are in opposite

half-planes, the pattern will follow sin (L, + L;) 0 and a rapid oscillation will result.

The oscillations in the analyzing powers described in ref. ‘I) and discussed later

show a period close to that of the I, = 0 (d, p) transition in fig. 1. Thus we will

be dealing exclusively with near-side/far-side interference rather than a Fresnel

pattern 25) arising from only one side of the nucleus.

In the case of the radial integrals from the adiabatic calculation, we would expect

to find that both near-side and far-side components are present, so a single-pole

3s R.C. Johnson et al, / Adiabatic distorted-d-waue approximation

form may prove inadequate. A number of dual-Pole f~rmuIae are possible; one that contains relatively few parameters is *3*29)

where L, is now complex and A is real and positive. The complex parameter L,

can be separated into two real parameters as Lz = Ly - iL:. Eq. (12) has an infinite number of second-order poles in the upper and lower half-planes. These lie at Lt+i(kvA-L:)and LF-i(krrA+L:)withk=1,2,3,..,oo_WhenL:>Oand L:<

=A, the pole with k = 1 is in the upper half”p~a~e and is the pole closest to the real axis. That pole will then give the largest contribution to the amplitude. When L: < 0 and Li> -TA, the pole with k = 1 is still nearest the real axis, but is in the lower half plane. Substitution of this form into eq. (3) yields both a far-side and a near-side amplitude,

&“I” = Bin:; = 9

(sin @)I’” exp (iL,R@)

exp (0) sinh (P-A@) ’

B, _, = Bj-?‘, = nn nn (sin H8),/’

(131

(14)

Because these two contributions appear for different values of X,, they contribute incoherently to the cross section. Their ratio, a measure of the relative sizes of the far-side and near-side cross sections, depends only on the parameter L: according to

(15)

An initial test of this model consists of a comparison of the form of eq. (12) with the simple 1, = 4 adiabatic zero-range calculation without spin-orbit distortions. To make this test, the 1, - 1, = 4 radial integrals were extracted from the calculations. The parameters of eq. (12) were adjusted in a least-squares sense to best reproduce the radial integrals for &&8. Although the integrals are not negligible for small values of 12, the contribution of partial waves with L2 < 8 to the cross section, when weighted by statistical factors, is in any case small. The results of the least squares fit are shown in fig. 5. The fitted radial integrals together with the model curves (dashed line) are plotted both as an Argand diagram and in modulus as a function of 1, value. The looping behavior near the maximum of the radial integrals is well reproduced. The values of the parameters are: R0 = 1.90+ 1.96i, Lz = 14.2 - 1.27i,

and A = 1.61. The value of L: is positive, and this radial integral gives a pre- dominantly far-side amplitude, as expected.

The analytic model just detailed makes predictions for both the near-side and far-side cross sections that are compared with the decomposition of the full calculation in fig. 6. The general slope and magnitude of the far-side piece are well


““Sn (d,pl”‘Sn U/2+) X,=4 Ed=79 MeV

39

t I I I I

-I 0 I 2

Re R f2 8n RI

Fig. 5. Comparison of the radial integral (see fig. 4) for 1, -1,=4 (open dots) with the multiple pole

representation of eq. (12) (solid dots). A representation is shown as an Argand diagram and in modulus

as a function of proton partial-wave number (l,).

reproduced, given that the model in no way can track small oscillations with angle.

For the near-side term, the values near 20” where the cross section is largest are

well reproduced; at larger angles the full calculation rises above this exponential

slope and the agreement worsens. Both near-side and far-side cross sections from

the calculation depend on the sums of irregular Legendre functions, and so are

unbounded at 0”. The rise toward this singularity is evident inside 5”. At these small

angles, a distinction between trajectories on the two sides of the nucleus is not in

any case physically meaningful. (When summed these amplitudes largely cancel,

giving a finite cross section.) In short, the semi-classical model appears to pick up

the essential features of the theoretically calculated cross section. The exponentially

growing ratio of far-side to near-side terms is expressed by eq. (15).

If, instead of the cross section, one examines directly the near-side and far-side

amplitudes from the calculation, two checks of the quality of the semi-classical


approximation are possible. The first reproduces the check of the slope of the far-side

cross section just made in fig. 6. Because the sech function is eq. (12) is squared,

the nearest and dominant pole is second order. If we approximate the amplitude

of eq. (12) by a single second-order pole at that location, it will generate an amplitude

given by

(sin/y2 B(+)cc exp (__gp) . (16)

Taking the far-side amplitude from the full zero-range calculation gives the angular

distribution in fig. 7 for I@‘)} ( sin 6)‘/‘/@. Over most of the angular range, the

amplitude falls exponentially. If we reproduce this with a simple exponential form

(dashed line in lig. 7), exp( - ye), we find y = 4.1. This compares favorably with

the value obtained from the radial integral fit of TA + Li = 3.79. In addition, the

quality of the semi-classical approximation may be judged by the extent to which

the full calculation in fig. 7 is a straight line.

A second comparison, possible at the level of the scattering amplitude, is to

examine the rate at which the phase of the amplitude advances with angle. The

single pole model leads us to expect a phase variation of the form exp (iLo). This

phase behavior is shown in fig. 8 and presented as the deduced L-value obtained

from a calculation made at 1” steps in 13. The full calculation shows a moderately

stable value over most of the angular range, indicating that the semi-classical model

with its fixed value of L captures the essential features of the calculation. The

deduced values can be compared with that obtained from the fit to the radial

integrals. The horizontal line in fig. 8, drawn at Lt-i-$, lies slightly above the average

IO2 t 1 I I I

= EXACT

0 l FORMULA

G4 - o\

0 ’ \ 0 \

0 \

o ‘1 0

1O-6- I I I m 0” 40” 80”

8 c.m.

Fig. 6. Comparison of the far-side (solid dots and lines) and near-side (open dots and dashed tine) cross

sections with the predictions of the multiple pole representation of eq. (12).


IO’

10;

lOi

10

ZERO RANGE

uz=-l/2

\ \

I I I I I I I I

30 60 90 e cm.

Fig. 7. Comparison of the dominant far-side scattering amplitude (with asymptotic quantum numbers

o, = -1, m, =$, and c2 = -4) plotted in the form of IBl(sin O)“‘/O with a straight (dashed) line. The

distorted-wave calculation is of zero range and omits deuteron and proton spin-orbit distortions.

8 cm.

Fig. 8. Comparison of the phase advance of the dominant, far-side scattering amplitude (presented as

the deduced L-value) with the size of LF+i determined from the multiple pole representation of the

I, - L, = 4 radial integral. The distorted wave calculation is zero range and omits deuteron and proton

spin-orbit distortions.

42 R.C. Johnson er al. ,I Adiabatic distorted-woue approximation

value from the full calculation. Part of this discrepancy originates from a systematic

rotation error between fitted and calculated radial integrals presented in fig. 5.

Without additional background terms, the model carries too few parameters to

reproduce this behavior more precisely.

The closeness with which the zero-range adiabatic transfer reaction calculation

is described by the semi-classical model provides encouragement in the use of this

approach. Beyond the particular choice of the functional form of the model, the

amplitudes themselves exhibit the exponential decline and the stability of phase

advance with angle expected from a system where a single pole carries all the

essential physical information. Despite some discrepancies in this example, we shall

find, when we consider the case with spin-dependent distortions included, that the

additional attraction provided for the deuteron and proton in the nuclear surface

by the spin-orbit potentials makes these features an even better approximation to

the semi-classical limit.

4. Implications of semi-classical limits for polarization observables

In this section the far-side dominance discussed in sect. 2 is shown to have

important consequences for the polarization observables. If we continue to neglect

spin-orbit distortion effects, then the observables are completely determined by a

single amplitude B!,,, in the limit of far-side dominance. This in turn leads to specific

relationships between normally independent observables. We will begin with a brief

review of this case. For any realistic problem, however, the spin-orbit distortions

are large and must be explicitly taken into account. These distortions create both

large differences among the amplitudes on the basis of spin projection quantum

number and also couple states of different spin projection in the deuteron channel.

These latter effects together with D-state terms which alter the np intrinsic spin

states from the value specified by the asymptotic deuteron projection vi, will be

referred to collectively as spin-flip contributions in the following discussion.

Despite such complications, much of the simplicity of the semi-classical model

remains, although certain of the relationships between observables do not hold

unless spin-flip terms are neglected. The size of the spin-flip amplitudes arising

from the deuteron spin-orbit interaction have been examined for 80 MeV deuteron

scattering 30) and appear to be small. Additional contributions to spin-flip will arise

from D-state effects and we will discuss those. Thus, in the fully spin-dependent

case the polarization observables carry new information, but far-side dominance

still imposes substantial restrictions. Only two large amplitudes remain that satisfy

the constraint of far-side dominance, and so a revised set of relationships among

the observables is obtained.

Quite generally, the (d, p) differential cross section can be divided into three

incoherent pieces, corresponding to the projection of the incident deuteron spin

along the quantizat~on axis, in this case the perpendicular to the reaction plane.

R.C. Johnson et al. / Adiabatic d~torted-wave approximation

For a spin-zero target we define the partial cross sections

The sum of these is then the usual differential cross section

63 =E>, +(3$+(%>_, .

43

(17)

(18)

Since the vector (A,) and tensor (A,,) analyzing powers depend upon operators

that are diagonal with respect to the axis normal to the scattering plane, these partial

cross sections can be written in terms of the differential cross section and two

analyzing powers as

(19)

From these definitions, the vector and tensor analyzing power are simply written

A = (do/dflR), - (da/dfl)L, Y du/dLl ’

A =(da/d~n),+(d5/d~n)-,-2(d5/d~n),

YY du/dO

(22)

(23)

We have seen that in the zero-range, no spin-orbit limit, far-side dominance

leaves us with only one large amplitude, B1,,_. The elements of the transition matrix

are then simply

Note that the relations in eqs. (18)-(23) are exactly true. If spin-flip contributions

are small, or zero as in the case of eq. (24), the asymptotic deuteron and proton

spin projections apply also at the point where the reaction takes place and this in

turn places strong restrictions on the relationship between the various magnetic

quantum numbers. In particular, the Clebsch-Gordan coefficients in eq. (24) guaran-

tee that non-vanishing transition matrix elements satisfy

m,=l,+u,-cr, (25)

as a consequence of angular momentum conse~ation in the no spin-orbit limit.


From eq. (24), the vector and tensor analyzing powers are now uniquely deter-

mined to be

A?>, = 0 , (26)

21, A,=-

3 z,+1 forjn=In+$, (27)

while

A,= -$, forjn=ln-f. (28)

The vanishing of the tensor analyzing power in fact depends only on the assumption

that spin-orbit distortions are ignored ‘4,24). The additional results for the vector

analyzing power in eqs. (27) and (28) require far-side dominance. Since the separ-

ation of the amplitudes into near-side and far-side components is not valid near O”,

there is no contradiction with the more general result, A,,(O") = 0. Zero-range distor-

ted wave calculations with no spin-orbit distortions agree very well away from 0”

with eqs. (27) and (28).

Transitions of the type j, = I, -i form a special case, since the far-side restriction

to 1, = A, also constrains the projection of the neutron total angular momentum.

No such constraint is present forj, = I, +$ transfer. In eq. (25), m, = I, -$ and hence

(+1 -a, = -4. The implications for the partial cross sections (already given in eqs.

(26) and (28) in terms of the analyzing powers) are

(29)

(30)

Fig. 9 compares the partial cross sections for the relationships predicted by eqs.

0 20 40 60 80 100

8 cm.

Fig. 9. Angular distributions for the three partial cross sections corresponding to the three projections

of the deuteron spin along the normal to the scattering plane. The distorted-wave calculations were made in zero range and include deuteron and proton spin-orbit distortions.

RC. Johnson et al. / Adiabatic distorted-wave approximation 45

(29) and (30) but for a zero-range calculation in which spin-orbit disto~ions are

included. Away from small angles, (dp/dfi), is substantially smaller than the other

two partial cross sections. We note that while (dc/dfi)_, and 2(da/dK?)0 are both

larger than (da/da)_, , they are definitely not equal. This leads us necessarily to a

discussion of the semi-classical results in the presence of strong spin-orbit forces.

Naturally, even in the presence of spin-orbit distortions, it is possible to continue

to divide the cross section into three components according to u, with respect to

the perpendicular to the reaction plane. However, the restriction CF~ - cr, = -1 for a

j,= a--$ transition holds only in so far as spin-flip terms are negligible. With the

inclusion of spin-orbit distortions, one must abandon the relationship of eq. (30),

since this compares the magnitudes of two partial cross sections and thus depends

directly on the assumption that there is no difference between the various spin

projection channels. The vanishing of (dc/dft), in eq. (29) required the additional

assumption of no spin-flip.

It is important to realize that even in the zero-range, no spin-orbit limit the results

reported here do not follow when j, = I, + + transitions are considered. In this case,

far-side dominance leaves two choices available for m,, and no constraints result

on the choices of crI and cr,, aside from the deuteron S-state stripping constraint

that ACT, - ~~1 s $. In an experimental sense, it is also worth noting that the most

serious problems occur for j, = 1, - $ transitions 11,‘4), thus we are focussed precisely

on the situation we wish to understand. An additional consideration is that the

choice of j, = 1, - 4 removes many of the amplitudes that might otherwise contribute

to the far-side part of the stripping cross section. In particular, the spin-orbit

favored ’ ‘) amplitude with (TV = - 1 and CF~ = 4 is excluded. Thus reduced, the far-side

component of the reaction does not completely obscure the effects of the near-side

piece which we will shortly discuss.

The vanishing of the (da/do), cross section in eq. (29) has two consequences

for the observables. By substitution into eq. (22)

A,cO, (31)

and by substitution into eq. (1 I), we also obtain

A,+3Ay+2=0. (32)

The quality of the agreement to these predictions is illustrated in fig. 10, which

shows angular distributions of the A,y and A, measurements for ““Sn(d, p)“‘Sn

to the :’ final state. The A, data is generally negative except at the largest angles

where the trend of the angular distribution would suggest positive values. The tensor

analyzing power measurements have been replotted as though they are vector

analyzing power measurements by using eq. (32) to generate & = -3(2 + A,,.y). These

measurements are shown by the open squares, and the difference between the two

sets is plotted at the bottom of fig. 10. The difference dearly does not vanish. The

difference is always positive since eq. (32) follows directly from the right-hand side


“6Sn(d,p) ‘17S~(7/2+) R ,=4 Ed=79 MeV 0.50 / I / I

0.25 - T

-Lo+ I.00 * AY

4 P &=-(2+A,,)/3

0.75- \

‘\ l \

0.50 ‘\‘\ ~

0.25: it&S

0.006 1 I l- -t 20 40 60 80

e c.m.

Fig. 10. Angular distributions of the vector analyzing power A,. (solid dots) and the tensor analyzing

power A,.,. recalculated for comparison as 2,. = -f(2+ A,., ) (open squares), for the ‘%n(d, p)“7Sn(p)

transition. The lower half of the figure shows the A, -A, difference. The distorted-wave calculations

which include (omit) stripping from the deuteron D-state are shown as solid (dashed) lines for each

angular distribution.

of eq. (1 l), itself a cross section and thus is positive definite. For values of &,. > 30”,

the S-state calculation for A, -A, is small. It is in this case that eq. (32) is confirmed

theoretically. (We will return later to a discussion of calculations including the

deuteron D-state as an example of a spin-flip contribution.) The observation men-

tioned in an earlier section, namely that interference patterns are present for the

analyzing power angular distributions, is clearly evident for this case in the neighbor-

hood of 30”. These patterns are remarkably similar in both A,, and &,, suggesting

a common physical origin. Their absence in the calculation points to a missing

near-side contribution to a theoreticat calculation already far-side dominated.

Knowing that u1 --a,= -f, it is possible to associate the (da/dfl)0 partial cross

section with spin up (c~=$) protons in the outgoing channel and (dc/dQ)_, with

spin down ((T* = -3) protons. Maintaining the approximation of eq. (29), the

outgoing proton polarization for unpolarized incident deuterons can be written

p = (da/dfin)o- (da/dfln>-, drr/dL? *

(33)

From this, two more relationships among the spin observables are obtained:

l-p+2A,=O, (34)

R.C. Johnson et al. / Ad~abaf~~ dist~rt@d-wake ff~~~o~irnat~~n 47

1 + 3p + 2A,>,_,z = 0 . (35)

Both of these relationships would be amenable to empirical testing if measure-

ments of the outgoing proton polarization were available. Such measurements are

underway for the 66Zn(d, p)67Zn reaction 31), in this case for an 1, = 3 transition.

Since this reaction connects the ground states of two stable nuclei, the proton

polarization can be measured as the analyzing power in the time-reversed (p, d)

reaction. Some initial results for eq. (35) are shown in ref. 14) and indicate that the

relationship, satisfied theoretically, fails experimentally.

Eq. (35) is special in the sense that it does not depend on the absence of spin-flip,

as suggested in the derivation just given. Following the argument in ref. 14), only

parity conservation, expressed through (-1)W2*mn-rrl*‘n = 1, and far-side dominance

in the sense that m, = I,, -4, are required. Spin-flip contributions may arise from a

number of sources, including deuteron-nucleus tensor forces and coupling to the

breakup channels as well as stripping from the deuteron D-state. The difference

between the solid and dashed curves in fig. 10 illustrates the effect of that part of

the spin-flip amplitude that arises from the D-state. The inclusion of this part is

essential for the forward-angle agreement with the tensor analyzing power measure-

ments and the associated large value there of the difference function near 0”. In the

Fig. 11.

0 20 40 60 80 100 8 cm.

Calculated angular distr~but~oos similar to those in fig. 10 for the outgoing proton p and the tensor analyzing power A,, recalculated for comparison as 5 = -f(ZA,,

polarization

+ 1).


middle range of angles it has little effect, and reappears again at large angles where

the momentum transfer is large. In no place away from 0” does it explain all of the

difference, even though half of the large angle difference may originate with this

amplitude. Nevertheless, the size of the spin-flip contribution from D-state stripping

leaves open the question of whether other spin-flip terms not included in our

calculation might make up the rest of the difference.

Fig. 11 shows the theoretical comparison between the outgoing polarization and

the tensor analyzing power. This time the tensor analyzing power is replotted as

though it were polarization through eq. (35), i.e. p” = -?j(l+2A,,). As in fig. 10,

calculations with and without the deuteron D-state are represented by solid and

dashed lines. The difference function, p-i, shows little change when the D-state

is included, indicating the expected insensitivity of this comparison to the presence

of spin-flip amplitudes. Since we do not yet have full transfer reaction calculations

containing tensor forces and D-wave breakup channels, the comparison of p and

p” shown in fig. 11 constitutes currently the best test, aside from the presence of an

interference pattern, for the idea of far-side dominance against experimental data.

The failure of eq. (35) in ref. 14) thus indicates that the problem with (d, p) reactions

extends beyond an underestimation of spin-flip effects.

5. The semi-classical model in the presence of spin

The extraction of the missing near-side component will rely upon a model capable

of reproducing the experimental measurements in the presence of large spin-orbit

effects. The availability of vector and tensor analyzing power angular distributions

for the “%n(d, p)“7Sn reaction provides the most complete set of information from

which to begin.

The measurement of the three partial cross sections, (dcr/dfi),, (da/da),, and

(dv/dfi)_, are shown in fig. 12, together with the full adiabatic transfer reaction

calculations. The solid curves include both the deuteron S- and D-states, while the

dashed curves illustrate the results of the S-state only. We expect, on the basis of

the semi-classical approximation, that away from 0” both (da/dL?), and (da/da)_,

should be larger than (da/do), . Beyond 20” in the former case and 10” in the latter

case, this appears to be true, except for the very largest scattering angles. Both

(dtr/dft)O and (da/do)_, show oscillation patterns that are not present at anywhere

near the required magnitude in the theoretical calculations. Closer inspection reveals

that the two patterns differ in detail in both their period and phase. These differences

reflect the spin-dependence of the interfering near-side amplitude. The semi-classical

approximation also suggests that (dg/dfl), is small. The data clearly exceeds the

magnitude of the calculation, and the difference is made up in part by the D-state

contributions. It is the vanishing of this partial cross section that leads to the test

illustrated in fig. 10, thus there is a similarity with the effects of the D-state in the

two plots. There is no interference pattern evident in (da/da), , so this partial cross

R.C. Johnson et al. / Adiabatic distorted-waue approximation 49

““Sn(d,pf ‘7sn(7/2+) Ed= 79 MeV

@i

20 40 60 60 100

8 c.m.

Fig. 12. Measurements of the three partial cross sections for the three deuteron spin projections along

the normal to the scattering plane. The distorted-wave calculations which include (omit) stripping from the deuteron D-state are shown as solid (dashed) lines for each angular distribution.

section will not be useful in making a first estimate of the size of the amplitude

required by the data. (We wilI return to this point again later.)

We now need to identify the way in which the large radial integrals contribute

to the amplitudes for the various partial cross sections. This information will allow

us to make a proper choice of function used to fit the amplitudes and hence model

the theoretical and experimental results.

Assuming only conservation of angular momentum, the transition matrix elements

for the (d, p) reaction on a spin-zero target can be written as 32)

(@2,_Lm,I if(d, p)la,)=C (Ml, la, Ij,m,)(l,h,,~~~lIj2,,)(j2m2,j,mnIj,m,)

x Y::x(&) YIZhZ(kp)R2jzjnl,;l s (36)

SO R.C. Johnson et al. / Adiabatic disforted-wad approximation

where the summation runs over I,, A,, j, , m ,, Z2, AZ, j,, and rn2. Eq. (36) is a general specification of the partial-wave matrix element, and represents the starting point for any theoretical model. In the case that the distorted wave adiabatic approximation is adequate, R can be expressed as a linear superposition of radial integrals corresponding to all possible values of the orbital angular momentum transfer I and spin transfer S. In the zero-range limit, R is proportional to the product of a 9j symbol with a radial integral similar to that described in eq. (4).

When I,, j,, l,, and j, are all large, the amplitude of eq. (36) can be significantly simplified. This has been carried out in detail by Ellis 33) and in a special case by Dean and Rowley 23), so we need only to sketch the procedure here. If we first chose the z-axis along kd [consistent with the Madison convention “‘)], all of the magnetic quantum numbers associated with f, , ji , 12, and j, are small. In this limit, simple asymptotic expressions for Clebsch-Gordan coefficients in terms of rotation matrices can be used. When, in addition, the standard asymptotic formulae are used for the spherical harmonics that appear in eq. (36), the resulting simple dependence of the matrix elements on CT,, pz, and rn, makes it possible to rotate the amplitude into the coordinate system in which spin projections are referred to the Madison y-axis (normai to the scattering plane). We can now generalize the separation of the matrix element into far-side and near-side pieces:

(37)

where

xs(j,,I,~~~)S(I~,/~f(CTZ+m,-cr,)),

and C” is a ~-independent constant.

(38)

It is useful at this point to note the similarity of eq. (38) with eq. (3). The assumption of large orbital angular momenta has again selected classes of radial integrals R as responsible for driving amplitudes with particular spin-projections. We note in particular that the combination cr, + m, - 5, has taken over the role of A, in determining I, - Z2. To make this connection more concrete, it is instructive to look at the relationship between the very general l? and the radial integrals R of eq. (3) in the zero-range limit. Again, when the entrance and exit channel orbital momenta are large, an asymptotic expression for the 9j coefficient in terms of Clebsch-Gordan coefficients can be used to give, for the particular combination appearing in eq. (38),

R.C. Johnson et ai. / ~diabaii~ di~iorted-wave approx~maiion 51

While spin-orbit distortion effects are significant we do not expect that they alter

the basic dynamical far-side matching arguments of the spin-independent case.

We thus fully expect that, in the case of the f*, 1, = 4 transition, for example,

those radial integrals I? with f, - Iz = 4 and consequently those amplitudes with

u2 + M, - crl = 4, will dominate. Thus, for each choice of the spin projections a, and

rr, we can associate near- and far-side projections of radial integrals I? of a particular

class, i.e. having a specific relationship among their quantum numbers. In the

particular case of j, = 5, I, = 4, those amplitudes and f? consistent with u2 + m, - o, =

4 are collected in table 2. The values for j, have been suppressed in the subscript

notation for clarity. The table is organized so as to make it clear that each partial

cross section is an incoherent sum of (at least) two terms that differ in the projection

of the final proton spin. For each of these terms, there is a series of I? radial integrals

that contribute to the far- and near-side pieces of each matrix element. That the

uz+ rn, - crl = 4 amplitudes considered above do in fact dominate is shown in

fig. 13. Here we compare the cr, + m, - crl = 4 squared amplitude I{ -$, $1 T/ - l)/“,

which contributes to (do/dfin)_,, with the o,-i- m,-cr, = 2 and 0 contributions

I(-+,$I?-I--l)l” and I(-$, -$1TI-1)12, respectively. This figure is the generalization

of fig. 3 to the spin-dependent case of CT, = -1 and a2 = -1.

It is clear from table 2 that the R’s that contribute to each matrix element are

different, as is indicated by a change in quantum numbers. Thus when we fit a given

partial cross section, at no point will we compare the far- and near-side projections

of the same radial integral. This means that any model for l? which contains poles

in both the upper and lower half-planes must necessarily carry indeterminate

information. For this reason, our model will utilize amplitudes that possess poles

TABLE 2

Values of the far-side and near-side R indices


IO5

g IO4 0 L ki IO3

% i2 0 IO2

N

x ; IO’ -

E’ -IN ,oo

4 -

16' 0 30 60

8 c.m.

Fig. 13. Angular distributions of the squared amplitudes (-f, m, 1 TI - 1) for CT,+ M, - CT, = 4,2, and 0.

in only one or the other half-plane. In fact, we will choose a form with only a single

pole.

Each radial integral appears in table 2 twice, once as a far-side projection and

once as a near-side projection. The symmetry of the spin coupling means that these

two projections are associated with the matrix elements whose proton and deuteron

spin projection quantum numbers in (uZ, jnmn 1 TI a,) are opposite. Thus the large

far-side projection of R1,,,2_1,2,1,+4,,,+3, associated with u, = -1, u2 = -f, has a near-

side projection contributing to the V, = 1, (T* = $ matrix element. At least in part, it

is this near-side projection that accounts for the non-zero value of the difference

function, A, -a,. (fig. 10) from the S-state calculation.

Also evident is that the partial cross sections contain contributions from amplitudes

other than the one we would specifically like to investigate. In particular, the partial

cross section (da/da)_, receives incoherent contributions not only from the large

( - $, $1 T 1 - 1) matrix element but also from ($, 2 1 T ( - 1). Thus we need some theoreti-

cal assurance from the start that these terms are at least small, so that they do not

become an unwanted background of significance in the extraction of the model

amplitudes. Fig. 14 shows the largest modulus squared matrix element for each

choice of (T, and uZ. These amplitudes, taken from the finite range S- and D-state

calculation, contain spin-orbit distortions and are labelled by (TV, v2, and m,. The

two largest terms away from 0” are associated with U, - m2= -4, namely u, = -1,

u2 = -4, and u, = 0, Us = f . This is just what is expected semi-classically if j, = I, - 4.

Except near 0” and at very large angles, all other contributions are smaller by at


I"'b I , i.( f-f-t,

f 1%

30 60 90

Fig. 14. Angular distributions of the largest squared amplitudes (02m, 1 Tla,} for each value of u, and

CI~. The actual quantum numbers, including m,,, are noted near the end of each curve.

least an order of magnitude. Likewise, if one investigates the size of terms with the same values of cur and cr, as those shown in fig. 14 but different values of m,, a similarly strong selection of the semi-classically favored terms appears.

The reaction dynamics that selects those radial integrals whose far-side projections drive the two largest matrix elements is sufheiently strong that, in the calculations, the matrix elements with all quantum numbers reversed are in practice dominated by the near-side projections of the same r?. The near- and far-side pieces of the matrix elements must in any case be equal at O”, as is evident in fig. 14.

6. Simple pole model representation of the experimental and theoretical polarized cross sections

If we adopt a single-pole form for the model amplitude such as that in eq. (7), there will be in general five parameters, including the order of the pole. Two interfering poles can have ten, but the overall phase is not determined in a fit to an experimental cross section, so the number of parameters is reduced to nine. In eq. (S), the real part of the pole position enters only in the phase of the amplitude as a function of scattering angle. As discussed following eq, (1 lf, when two poles interfere, only the sum of the real parts enters, and another parameter is lost. We can only recover the absolute magnitude of the real pole positions by introducing further information, in this case from the theoretical calculations of the far-side amplitude. Thus each partial cross section (i= vt =O, fl) will be represented in

54 R.C. Johnson et al. J Adiabatic distorted-waue approximation

our model by a function of the form

#(‘*-I~ =A”-e

sin 0

+2A8”-‘8”-’

sin 0 e-(“+‘“)“‘2[a cos A@+ 5 sin A@] . (40)

Here the parameter A, chosen to be real, and the complex parameter A’ = a + id are

the amplitudes of the two pole terms of the model. The distances of the poles from

the real axis are given by the slope parameters $r and ;I-‘. The orders of the poles

are p and Y and the sum of the real axis pole positions is ~4. A priori, there is no

way to identify from the fit which amplitude is far-side and which is near-side. For

this, we will again turn to the theoretical computation of the far-side amplitude,

and associate the far-side amplitude with the fitted pole term whose slope it matches

most closely.

The measurements for (dcr/dft), and (da/d&?)-, were reproduced by the formula

of eq. (40) in which the parameters were assigned so as to minimize x2. The orders

of the poles were assumed to be discrete and were chosen ahead of the fit, leaving

six adjustable parameters. As we later wish to make a correspondence between part

of the fitted amplitude and the theoretical far-side amplitude, the order of the pole

was set to second order (p = 2). This is the order most appropriate for the theoretical

calculation, as we shall demonstrate Iater. For the (drr/dtin), partial cross section

this was in fact the only choice which led to a convergent fit. No sensitivity was

found in the fits to the order of the pole later to be identified with the near-side

amplitude, so for consistency we took Y = 2. The final parameters that best repro-

duced the (du/dG), partial cross section are given in table 3, together with errors

taken from the diagonal elements of the error matrix from the fitting process.

In fitting the (da/dR)._, partial cross section, there was difficulty reproducing

the measurements between 6” and 10” because of the large size of the near-side

amplitude required by the interference pattern. As the physical separation and

interpretation of near- and far-side amplitudes is in any case suspect at such small

angles, these points were discarded. The parameters which fit the oj = -I partial

cross section are shown in table 3 along with their errors. These errors have been

increased by a factor of (x/u)“‘, where X/Y is the reduced chi square, to take

account of the imperfect fit, The reproduction of (dcr/dfi)_, , shown in fig. 15, is

however satisfactory beyond 13”.

The identification of one of the fitted poles as an upper half-plane pole (far-side

amplitude) is made by reference to the far-side amplitudes from the theoretical

calculation, The calculated far-side projection of the transition amplitude with

(ut, %, m,)=(-1, -~,I),i.e.f=(-t,~lT’+‘I-l), q re uired for (du/dfi)_, is plotted

in the form /f/(sin S)““/Q in fig. 16. From the amplitude we can deduce a slope y

for the form eWy” close to 4.21. This is in excellent agreement with $r in table 3


TABLET

Double-resonance parameters

55

(dc+/dfl),

A 35.36*0.80

a 3.5*1.2

a O.O* 1.6

*r 3.549*0.035 $1 4.05 + 0.61

A 26.04 * 0.75

x21 * 1.7

W/da)-,

57.8zt4.7

219* 121

-32*67

4.3650.13 11.1* 1.5

20.0 + 1 .o

4.7

and identifies the pole of strength A with the theoretical far-side amplitude even

though, according to table 3, this not the pole with the largest coefficient.

Having identified which of the fitted pole terms is to be associated with the

calculated far-side amplitude, we can plot the two pole terms separately to obtain

a first impression of the angular distribution of the near-side amplitude required

by the data. Fig. 17 shows the far-side (solid lines) and near-side (dashed lines)

amplitudes. A band is noted in each case indicating the magnitude of the error

estimated from the fitting process. The large errors on the parameters a and d reflect

errors in phase and slope as well as overall magnitude. Thus, in this situation it is

more representative to consider the full error matrix when calculating the errors as

a function of scattering angle. It is clear that the near-side amplitude is best

determined in the angular region near 20” where the interference pattern is most

pronounced. The oscillations are sufficiently large there that the fitted near-side and

Fig. 15.

“%(d,p) “‘Sn (7/2+) Ed=79 MeV

8 cm. Angular distribution of the (da/do)_, partial cross section and the model reproduction

upon eq. (40).

based

R.C. ~0~~~0~ et al. / Adiabatic d~torted-waft approximation

IO-

'0 t I / ,

20 40 60 80 I9 c.m.

Fig. 16. Angular distribution of the far-side piece of the (-f.31 TI- 1) amplitude f plotted as

[fi(sin 0)“2/0. The exponential which best reproduces this amplitude is shown by the dashed line.

the theoretical far-side amplitudes are of the same size. Since the interference pattern

dies quickly with angle, there is a steep slope to the near-side amplitude, reflected

in a large value of $‘. This makes the near-side amplitude exceed the far-side at

small angles. It is the magnitude at small angles that governs the size of the pole

strength parameters A and A’ in the fit, and makes /A’/ > A in this case. In this

sense, what we are missing is bigger than what we have. Both amplitudes show the

roll-over expected near 0” due to the choice of p = Y = 2. At large scattering angles,

the interference pattern has disappeared and only an upper limit really exists for

the near-side amplitude. The error band represents an extrapolation of the slope

a INDIVIDUAL AMPLITUDES FOR (du/diH_, IO j- I I I I I I I 1 I

\ \ 104- \ \ \ -

\ \ \ , I , I \I I I

20 40 60 80 8 cm.

Fig. 17. Near-side and far-side amplitudes contributing to the pole model for (do/da)_, . Fitting errors

indicate a range of values for each amplitude whose limits are shown in each case by the pair of lines.


determined at smaller angles using the single pole form of the amplitude. It is thus

likely that the actual near-side amplitude is bigger at large angles than is indicated

in fig. 17.

Additional information regarding the position of the real part of the near-side

pole can be obtained from the theoretical far-side pole position. The phase advance

ofthe(-:,?ITI-l) amplitude, interpreted as an effective L-value, is shown in fig.

18. As mentioned earlier, this quantity is even more stable with angle than in the

similar plot, shown in fig. 8, from the calculation without spin. The average L-value

at angles beyond 15” is 12.5. With A = 20.0 from the experimental fit, we obtain a

real position of l2 = 7.5 for the near-side pole of the U, = -1 partial cross section.

We note that the phase advance plot reaches its stable (semi-classical) value only

at about 15”, so, if the experimental cross section is sensitive to the same physical

result, the above mentioned inability of the semi-classical model to reproduce

partial cross sections inside 13” is understood. On the basis of the radial integrals

in table 2, it is reasonable to expect that the far-side projection of the empirical

amplitude, whose large near-side projection is shown in fig. 17, should account for

the difference between the theoretical and experimental values of (da/da), evident

in the top panel of fig. 12. If spin-flip terms, other than those from the D-state, are

not large, this difference may be regarded as an upper limit on the far-side projection

of &,+1,2,1,-4.1*~3 * In a similar way, the w, = 0 partial cross section was analyzed. The model fit to

all the data is shown in fig. 19 and the deduced pole parameters are included in

table 3. The slope parameter of the calculated far-side projection of the ($, ;I TI 0)

amplitude deduced from fig. 20 is close to 3.16, which once again identifies the pole

of strength A with the theoretical far-side amplitude. Fig. 21 shows the angular

14 -

IO - AVERAGE FOR &,,.>I50 -

L 8-

PHASE ADVANCE FOR

<-:,$ ITi -I&

!

I I I I I I I

0 20 40 60 80 8 c.m.

Fig. 18. Phase advance for the far-side piece of the (-f ,;I TI - 1) amplitude. The average value for all angles past 15” is indicated by the dashed line.

58 R.C. Johnson et al. / Adiabatic distorted-wave approximafion

“6Sn(d,p)“7Sn(~Z+) 1000~

Ed =79 MeV 8 I I /

I I I I I 1 I 20 40 60 80

8 cm.

Fig. 19. Angular distribution of the (do/do), partial cross section and the model reproduction based upon eq. (40).

distribution of the far-side (solid lines) and near-side (dashed lines) amplitudes

required by the data. As in the (To = -1 case, the sets of double lines indicate the

error band. The near-side amplitude is determined by the interference pattern

between 10” and 40” which, unlike the (T, = -1 case, extends to larger angles. This

is reflected in the fitted pole parameters through comparable values for ;I’ and @’

and in fig. 21 by near- and far-side amplitudes which run nearly parallel.

The position of the real part of the near-side pole is again obtained from the

theoretical far-side pole position. The phase advance of the (4, ;I T]O) amplitude is

IC

'0 t I I I

20 40 60 80 e cm.

Fig. 20. Angular distribution of the far-side piece of the (f, 3 1 TIO) amplitude f plotted as }f/(sin 0)“2/@.

The exponential which best reproduces this amplitude is shown as the dashed line.


IO’ INDIVIDUAL AMPLITUDES FOR (d~/d~)O

/ I , I f I I I

10-j -

lo-4 -

lo‘5o 1 I I , I I 20 40 60 80

8 c.m.

Fig. 21. Near-side and far-side amplitudes contributing to the pole model for (drr/dR),

shown in fig. 22. The average value of L at angles beyond 20” is 13.2. With A = 26.04

from the fit, we deduce the real pole position, 12= 12.8, for the near-side pole of

the (T, = 0 partial cross section.

Thus our simple pole model has revealed that the data require large amplitudes

and the period of the observed interference patterns dictates that these amplitudes

arise from the lower half-plane (near-side) poles. In addition to their large size,

these near-side projections of the amplitudes show a very strong spin dependence.

This is evident not only in the different rates of fall-off of the w1 = -1 and o, = 0

amplitudes with scattering angle, but also in the real-axis positions of their poles.

14

12

IO

L 8

6

4

2

I 0 0

:/ AVERAGE FOR 8c,,,>200

PHASE ADVANCE FOR

<$,;/T\O)tar

J I I I I 1 I /

20 40 60 80

9 c. m.

Fig. 22. Phase advance for the far-side piece of the (4, $1 TIO) amplitude. The average value for all angles

past 20” is indicated by the dashed line.

60 R.C. Johnson et al. / Ad~a~~i~c distorted-wade apprnx~mat~on

For (da/da)*, the real parts of the upper and lower half plane poles are nearly

the same, indicating that both contributions are surface-peaked. For (da/dLJ).., ,

the partial wave associated with the near-side pole is half that of the far-side pole.

Since the far-side pole is again associated with grazing partial waes, the near-side

amplitude in this case clearly arises from the nuclear interior. It is perhaps significant

therefore that the amplitude (-4, %I T/ - I) d riving (dcr/dti)_, has the proton spin

projection which, on the near-side of the nucleus, results in a large attractive potential

due to the spin-orbit force.

7. Character of the new amplitude and its impIications

Our semi-classical analysis has shown that the bulk of the physical content of an

adiabatic breakup calculation in medium energy transfer reactions can be success-

fully modelled using a simple pole in I-space with a pole position that corresponds

to flux travelling around the far side of the nucleus. While no evidence has been

found that this far-side flux is incorrectly described by the theory, the separation

of the amplitude into far-side and near-side pieces has revealed that near-side

amplitudes required by the experimental data are absent from the theoretical

calculation. The primary evidence for this is the presence of an interference pattern

in the analyzing power, or alternatively (da,/dQ& and (dcr/dfl)_, , angular distribu-

tions whose period of oscillation corresponds to the diameter of the nucleus. In this

section we will review the quantitative information obtained from our analysis that

constrains the properties of this missing amplitude.

The description of the 1, = 0 badly-matched ground-state transition in

“%(d, p)“‘Sn appears to be fairly satisfactory. Both far-side and near-side ampfi-

tudes arise from the existing radial integrals, and no serious problems are encoun-

tered in the description of the cross section and analyzing power angular distribu-

tions. As the I, transfer rises, the far-side amplitude increases with improvements

in the linear and angular momentum matching. This matching results in a relative

suppression of the near-side components of the amplitude. Yet we find that the

measurements require near-side components comparable with those from the far

side; thus they too must rise as the matching on the far side improves.

This raises the question of the momenta involved in both processes. Simple

estimates can be made if we assume that the reaction is sufficiently surface peaked

that we may neglect the focal changes due to the action of the optical potentials.

Asymptotically, the deuteron and proton momenta are roughly kd - 2.7 fm-’ and

k, - 2.0 fm-*. Matching on the far side, shown in fig. 23, gives a neutron momentum

of k, - 0.7 fm”“‘, which agrees very well with the linear momentum expected for an

1, = 4 neutron travelling at the surface of an optical potential of mean radius 5.X fm.

It is in this sense that the transition is well matched. At the point of transfer, the

neutron-proton relative momentum in their own center-of-mass is approximately

$( k, - k,), or 0.65 fm-‘. This value is well within the range of momenta where the


d

61

Fig. 23. Simple trajectory diagrams for the two interfering amplitudes for either (du/dlt), or (da/da)_.,

The top diagram (a) is for the far side and the bottom diagram (b) is for the near side.

formfactor Vn,Cpd is large, and overlap between the entrance and exit channels is

good.

In the near-side amplitudes that interfere with the large far-side contributions in

(da/do), and (dcr/dL!_,, the neutron and proton must separate back-to-back,

also shown schematically in fig. 23. The neutron is stripped into an orbit rotating

in the same sense as that generated by the far-side amplitude and the proton proceeds

directly toward the detector along the near side. In this case the neutron-proton

relative momentum is larger; i(k,- k,) = 1.35 fm-‘. At these momenta the S-wave

part of Vnp& is falling, and the D-wave part is approaching its maximum. Thus

we might reconsider the amplitude structure in search of evidence that the deuteron

D-state, through its emphasis on higher momenta, can contribute to the resolution

of the difficulties with the reaction mechanism.

A comparison of the calculations with and without the deuteron D-state reveals

only one amplitude that shows both a large increase when the D-state is included,

and a large enough magnitude to make a significant contribution to the reaction

process, namely ( -1, II TI 1). This is shown in fig. 24 in both the S-state only and

the S + D-state form. Its effects are seen in (da/da), as a result of the (+1 = 1 deuteron

projection quantum number, and by the same token it also appears in the angular

distribution of A, -A_” displayed at the bottom of fig. 10. The break in the slope at

40” marks the point where it ceases to fall at the same rate as most other amplitudes

B c.m.

Fig. 24. Angular distributjon of the squared amplitude (i.4, $1 T/l) with (S+ D) and without (S only)

stripping from the deuteron D-state included.

in the reaction (see fig. 14). Thus its relative contributions rise with angle, as can be easily seen in the large angle portion of the A, -& difference in fig. IO. The quantum numbers of o, = 1 and cr, = -4 indicate that this amplitude falls within our classification as a spin-flip process, which is consistent with that part of the stripping from the deuteron D-state that proceeds with spin transfer 2. It remains large at large angles primarily through the action of the deuteron spin-orbit force for the o1 = 1 projection [rather than the strong proton spin-orbit effect that drives the (d, p) reaction j-dependence ‘“)I. Elastic deuteron scattering in this spin state shows a diffraction pattern (originating from a single side of the nucleus) with a (rainbow) maximum near 40” and a diffractive minimum near 30” [ref. ‘“)]. This pattern appears again in this amplitude for the (d, p) reaction. (It is thus apparent that much of the difference in A,, - .&, could be explained if other processes, such as stripping from breakup D-states, could be added to the present model to increase the strength of the spin-flip contributions.) This amplitude, however, does not contribute to either (da/df2)* or (da/d,C1)_, , nor does it affect the py -$,, difference. Thus it is not the resolution to all of the reaction difI&lties, and in particular will not help with the explanation of the interference patterns represented by our model.

No other amplitudes, especially those with the quantum numbers of (4, f 1 TIO) and ( - f , $ / T I- I}, showed similarly promising effects. One explanation may lie in our use of the adiabatic approximation for S-wave breakup in the present calculation.

R.C. Johnson et al. / Adiabatic disrorjed-wad approximation 63

The model explicitly assumes that breakup exists only for low-energy relation np

configurations. Additionally, it takes no account of any local or asymptotic changes

in the np center-of-mass energy associated with the energy needed to couple to the

unbound np channels. This may involve energies as high as 20-30 MeV [ref. ‘)I.

Since it is precisely these high-energy np configurations that the near-side amplitude

would like to emphasize through the larger np relative momentum at transfer, a

major contribution may be missing because of the incorrect dynamics of the np

relative and center-of-mass motion. This would indicate the need to replace the

adiabatic approximation with a fully coupled channels (CDCC) calculation, referred

to in the introduction, which includes S-wave and D-wave breakup components.

The semi-classical model also provides us with the orbital angular momentum at

which most of the flux appears in each spin channel. In the case of (da/dQ)o, the

far-side amplitude is largest for i2 = 13.2. The radial integrals of table 2 allow us to

estimate the peak orbital angular momentum in the entrance channel to be I, = I, + 4 =

17.2. At these energies, these are grazing orbital angular momenta in both channels.

For the near side, the sense of coupling reverses. Beginning with Z2 = 12.8, we obtain

an entrance channel value of I, = 8.8, representing a substantially more central

collision. In the (da/do)_, channel, the situation is even more extreme. The far-side

amplitude has l2 = 12.5 and i, = 16.5; again both are grazing values. But the near-side

amplitude has its largest outgoing channel value at &=7.5, forcing 1, =3.5, a very

small value. This depicts a truly central collision between the deuteron and the

nucleus.

The enormous differences between orbital angular momentum numbers for

(da/d0)o and (da/do)_, is evidence for a very large spin dependence in the

anomalous near-side term. Differences in the optical potential geometry due to

spin-orbit effects alone amount to radius changes of the order of 0.1-0.2 fm, changes

similar to the differences in l2 obtained from the far-side amplitudes in the two spin

states. The changes in the anomalous term are well over an order of magnitude

larger, even given the uncertainties in the estimate.

A similarly large spin dependence appears in the angular distributions themselves.

As we learned from the semi-classical model analysis, the slope of the cross section

angular distribution is related to the width parameter of the generating pole in

I-space, with slower descents corresponding to poles closer to the real axis. Thus

the slope of the far-side amplitude represents a width of a few units in orbital

angular momentum, and characterizes what we mean by a surface peaked reaction.

For (dv/da),,, the new near-side amplitude has almost the same slope, and is

therefore surface-peaked as well, a result at least consistent with the grazing values

for outgoing proton momentum. At the same time, the (da/do)_, term falls sharply

with angle. This corresponds to a very broad spectrum of contributing I-values.

Thus the flux in this case can almost be said to originate uniformly from the surface

of the nucleus exposed to the detector. The slope is su~ciently steep, and the range

of t-values for signi~cant radial integrals sufficiently large, that unphysically large

64 R.C. Johnson et al. ,I Adiabatic distorted-wave approximation

components (well outside the nucleus) may be implied by our simple pole model.

However, removal of these terms from this amplitude will disturb the predictions

only where the near-side amplitude is smallest. At large angles, the values shown

in fig. 20 are not determined from the experimental data at those angles, since the

interference pattern has vanished by then. Instead, they represent an extrapolation

from forward angles on the basis of the pole structure. At large angles, this pole

model probably cannot be maintained. Nevertheless, the near-side amplitude there

is at least an order of magnitude smaller than that for the far side, and the details

of its large angle distribution are of no consequence to our conclusions.

Given the large size and different dynamics involved in the missing near-side

amplitude, it appears that we are dealing with a process that is not contained within

the adiabatic distorted-wave model. Thus we do not see how minor adjustments to

optical potential parameters and the representation of the neutron bound state, such

that they remain within customary bounds, can make a sufficient change in the

reaction dynamics to destroy the strong far-side dominance. There is hope that

additional spin-flip amplitudes will prove helpful. These can originate in the fully

coupled-channel breakup calculations (CDCC), but at present the ability to include

all the important ingredients of spin-orbit forces and the deuteron D-state are not

at hand. Nevertheless, we feel that these calculations offer the best hope for a better

model of intermediate-energy transfer reactions.

8. Conclusions

We find that at medium energies (100 MeV), the (d, p) stripping reaction exhibits

semi-classical features that may be understood from the point of view of localized

trajectories characteristic of the effects of attractive forces in the nuclear surface.

We have used these features in making a near-side/far-side separation of the reaction

amplitude, and find that theoretically the far-side amplitude is dominant (making

the reaction coplanar). This leads to simple semi-classical features in the large

far-side amplitudes, including a smooth exponential behavior and a steady phase

advance with increasing scattering angle. The rate of fall of these amplitudes with

angle is consistent with the expected surface-peaked nature of the reaction.

Far-side dominance also constrains the number of spin-channel matrix elements

that can be large, and leads to redundancies among the spin observables. These

relations differ in their sensitivity to spin-flip in the deuteron channel, and all are

easily tested experimentally. At present, the experiments do not support them,

indicating that some of these semi-classical features, satisfied to a high degree of

accuracy by current theoretical calculations, are seriously in error. Angular distribu-

tions of the analyzing powers show interference patterns which indicate that the

property of far-side dominance cannot be maintained.

It is straightforward to model accurately the semi-classical features of the present

theory with simple poles in orbital angular momentum space. Using such a model


together with information deduced from the existing calculations, we can make an

estimate of the size and character of the missing amplitudes. We find that there

must be near-side pieces describing nearly central deuteron-nucleus collisions. Large

neutron-proton momenta are involved and an enormous spin dependence is

apparent. Using the currently best available reaction model, we do not see a means,

by variation of input, to change the far-side dominant features to an extent that

these amplitudes will arise theoretically.

There is in new initiatives, such as full discretized continuum (CDCC) calcula-

tions, the prospect of sufficient dynamical changes to make an explanation of this

process possible within the established three-body framework. The nearly central

collision characteristic of the missing near-side amplitude gives rise to neutron-

proton separations with opposite senses of rotation about the nucleus. These large

np momenta can only arise at the expense of a substantial reduction in the np

center-of-mass energy, which the adiabatic prescription assumes is unchanged. Such

differences are within the scope of CDCC calculations to reproduce, thus providing

optimism that such calculations may contain the large near-side amplitudes charac-

terized by this work.

We wish to thank B.K. Park for assistance with the calculations shown in this

paper. The financial support of the Science and Engineering Research Council (UK)

(for R.C.J. and J.A.T.) and the National Science Foundation (US) (for E.J.S.) which

made this collaboration possible is gratefully acknowledged.

Note added in proof: The cross section scales of figs. 4 and 6 of ref. “) and fig. 1

of ref. “) should be kb/sr rather than mb/sr.

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Nature of the amplitudes missing from adiabatic distorted-wave models of medium energy (d, p) and...

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