This research was sponsored by the National Science Foundation Grant NSF PHY 87-1440

Johnson, R.C.

Stephenson, E.J.

Tostevin, J.A.

English

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4. F.G. Perey and B. Buck, Nucl. Phys. 32, 353 (1962). 
5. E.J. Stephenson, R.C. Johnson, J.A. Tostevin, V.R. Cupps, J.D. Brown, C.C. Foster, 
J.A. Gering, W.P. Jones, D.A. Low, D. W. Miller, H. Nann, C. Olmer, A.K. Opper, 
P. Schwandt, J.W. Seubert, and S.W. Wissink, Phys. Lett. B 171, 358 (1986). 
6. E.J. Stephenson, R.C. Johnson, and J.A. Tostevin, contribution to this report. 
NATURE OF THE MISSING NEAR-SIDE AMPLITUDE IN CALCULATIONS 
OF INTERMEDIATE ENERGY (d,p) AND (p,d) REACTIONS 
E.J. Stephenson 
Indiana University Cyclotron Facaeilit y, Bloomington, Indiana 4 7405 
R.C. Johnson and J.A. Tostevin 
University of Surrey, Guildford, Surrey G U2 5XH, United Kingdom 
Model calculations of the cross section and analyzing powers for (d,p) and (p,d) trans- 
fer reactions at energies near 100 MeV often bear little resemblance to the measured 
angular distributions. To provide a basis for a detailed investigation, cross section and 
analyzing power angular distributions have been made for two time-reverse pairs of re- 
actions, 116Sn(d,p) '17sn with 117~n(p,d) l16sn (Refs. 1 and 2) and 6 6 ~ n ( d , p ) 6 7 ~ n  and 
6 7 ~ n ( p , d ) 6 6 ~ n  (presented elsewhere in this report). These studies find the greatest prob- 
lems for j, = l ,  - $ transitions, where the presence of a marked interference pattern 
in the analyzing power angular distributions indicates nearly equal contributions to the 
reaction amplitude from the far and near sides of the nucleus.* In distorted wave Born 
approximation calculations there is almost no near-side contribution, and the model an- 
gular distributions show almost no interference pattern. We have continued to investigate 
this issue using semi-classical reaction analysis techniques with the intention of extracting 
a phenomenological extimate of the size and character of the missing near-side amplitude. 
The data for this investigation come from the l ,  = 4, j, = $ transition in " ' ~ n ( d , ~ )  '17sn. 
A semi-classical analysis of the reaction amplitudes may be pursued in this case for 
two reasons. First, the typical deuteron and proton partial waves that contribute to 
the stripping or pick-up amplitude are large enough that semi-classical approximations 
are a useful representation of the reaction. Second, the dynamics of angular momentum 
matching at the nuclear surface have sufficiently strong effects that only a few of the 
possible amplitudes contribute significantly to the reaction. This leads in the case of the 
model calculations to redundancy relations among the polarization observables which are 
not matched by experiment.4 
In Fig. la ,  this dynamic selectivity picks out one projection (A,) of the transferred 
neutron's orbital angular momentum. In a model calculation (without spin-orbit distor- 
t ions for simplicity), this is the maximal projection normal to the asymptotic reaction 
plane with the sense of rotation commensurate with far-side scattering. Since this is a 
Figure 1. (a) Angular distributions of the absolute squares of the reaction amplitudes 
for the five allowed projections of the transferred angular momentum, An =4, 2, 0, -2, 
and -4. This model calculation is zero range (no deuteron D-state) and has no spin- 
orbit distortions. (b) Angular distributions of the absolute squares of reaction amplitudes 
where A, = 4. The "no spin-orbit" case is repeated from Fig. la ,  the new curves with 
spin-orbit distortions correspond to deuteron and proton projection quantum numbers of 
(01,02) = ( 0 , i )  and (-I,-$). 
jn = en - transition, there is likewise only one projection (m,) for the total angular 
momentum (j,) for which the amplitude is large, the maximal one (m, = 712). If we 
quantize along an axis normal to the plane, the Bohr rule requires that the deuteron and 
proton spin projections (01 and 02) obey the relation 
Thus only two paris of values, (al,oz) = (0, i) and (-1, -i), are allowed for the spin 
projection quantum numbers. Fig. l b  shows that even with spin-orbit distortions, both of 
these amplitudes remain large. Their difference attests to the strength of the spin-orbit 
distort ions. 
By looking at the partial reaction cross sections for each projection of the deuteron 
spin, we may select these amplitudes individually for study. Fig. 2 shows measurements 
corresponding to the three partial cross sections 
along with full model calculations. These calculations include adiabatic deuteron distorted 
waves, exact finite range with the deuteron D-state, and non-locality corrections. The po- 
tential parameters are taken from Ref. l. Both ( d ~ / d R ) ~  and ( d ~ / d O ) - ~  show interference 
oscillations not present in the calculations, and (doIda) 1 is substantially underestimated. 
Figure 2. Angular distributions of the 
three deuteron spin projection partial 
cross sections compared with full 
model calculations. 
Since the model amplitudes represent a strongly surface-peaked reaction, they may 
be replaced in a semi-classical approximation by a resonance in partial wave number. For 
this purpose we chose the form 
which is written as a resonance in terms of the proton partial wave number (12) For 
simplicity in this investigation, the resonance parameters (Ro, L, and A) were adjusted 
to reproduce the &-dependence of the largest radial integral contributing to each partial 
cross section. The spin coupling for each obeyed the constraint implied by Eg. (I), and 
was: 
The parameters for these fits are given in Table I. The reproduction of the surface resonance 
for both cases is shown in Fig. 3. 
Figure 3. Argand diagrams showing the amplituded associated with the ol = 0 partial 
cross section (a) and the = -1 partial cross section (b). The solid line connects values 
associated with the best-fit resonance formula. 
Table I: Parameters for the Semi- Classical Model Resonances. 
(doldn) 0 (doldn) - 1 
A near-asidelfar-side decomposition of the radial integrals in Eqs. (4) and (5) shows 
that their contributions to ( d ~ l d f l ) ~ ,  and (doldn) - 1, comes only from their far-side pieces. 
(For this to be strictly true for ( d ~ l d b l ) ~ ,  we must in addition assume that the dominant 
amplitude is (ozM I T I al) = (i, $ I T I o).) For an interference pattern to exist, a near- 
side amplitude of comparable size must exist. Information concerning this ar r 1 pl itude comes 
from the quantum numbers of that amplitude and the sizes obtained form adjusting it to 
reproduce the measurements. This latter analysis assumes that the partial cross sections 
are the result of interference between two surface-peaked resonances. The measurements of 
( d ~ l d b l ) ~ ,  and ( d ~ / d n ) - ~ ,  were reproduced by a formula involving two interfering single- 
pole resonances. In such a fit, the near-side and far-side character of the resonances is lost. 
Afterward, the resonance whose parameters match the model resonance is assumed to be 
far-side, and the remaining set is associated with the missing near-side amplitude. 
The form of Eq. (3) contains both near-side and far-side pieces (as it must to reproduce 
a real model radial integral). In the fit, we used a single-pole form to represent the pole 
of Eq. (3) closest to the real axis 
The form used to reproduce ( d ~ l d n ) ~ ,  and ( d ~ / d n ) - ~ ,  was 
where an overall phase has been removed, making A real. The orders of the two poles are 
m and n, their widths are a and p, and the period of interference pattern is governed by 
t = L(1) + L(2). The nearest pole of Eq. (3) is second-order (n = 2). Fits were attempted 
for all orders (values of m and n). Starting values were chosen so that a > P, thus insuring 
that most of the large angle cross section (ascribed to the far-side amplitude) would be 
reproduced by the first term in Eq. (7). A sensitivity to order was observed only for the 
first term, with ( d ~ / d R ) ~  preferring m = 2 and ( d ~ / d n ) - ~  preferring m = 1. To obtain 
a useful comparison with the resonance parameters of Eq. (3), we chose the fit where 
m = n = 2 in both cases. These fits are shown in Fig. 4 and the parameters of Eq. (7) 
listed in Table 11. The correspondence of the two resonant forms Eqs. (3) and (6) requires 
Figure 4. Measurements of the ol = 0 
and ol = -1 partial cross sections, to- 
gether with the fitted double-resonance 
formula shown over the angular range 
where data was included. The forward 
angle ol = -1 points were omitted. 
Table 11. Double Resonance Parameters. 
( d o l d n )  0 ( d a l d n )  - l 
where LI is the imaginary part of L. The comparison of these quantities for the resonance 
underlying the first term of Eq. (7) is given in Table I, and supports the association of 
these amplitudes. 
While both near-side and far-side pieces are needed to fully describe the amplitude 
missing from the model calculations, we may examine the size of the contribution from 
the single near-side pole of Eq. (7) assuming that it represents the bulk of the ampli- 
tude. For the ( d ~ / d n ) - ~  case, the far-side portion of the missing amplitude will con- 
tribute to (da/dn)l. Assuming that all of the difference between the model calculation 
and the data for (doIda) 1 is attributed to this term, an additional contribution of only 
10% is obtained. Figure 5 shows Argand plots of the two model amplitudes described by 
Figure 5. Argand plots of the model am- 
plitudes and missing near-side amplitudes 
for ( d ~ / d n ) ~  and (da/dR) The num- 
ber denote partial wave in the proton 
channel. 
Eq. (3). In addition, the missing near-side resonances of Eq. (7) are shown. For ( d ~ l d n ) ~  
the additional piece is small while for ( d ~ / d n ) - ~  it is substantially larger than the model 
amplitude. The size of these terms rests entirely on the a and a coefficients of Table 11. 
since it is only the size of the interference pattern that determines these coefficients, large 
uncertainties result. In addition, the missing amplitude for ( d ~ / d n ) - ~  is large over a 
broad range of partial waves, and these tend to cancel in the sum leading to the calculated 
cross section. Despite these uncertainties, it is clear that a near-side amplitude comparable 
in size to the far-side amplitude from the model can be generated that will explain the 
interference pattern. 
Additional information concerning the missing, near-side dominated amplitude may 
be obtained from the spin coupling for its associated matrix element. For interfererice to 
occur in the (daldn) - case, where this amplitude is the largest, the radial matrix element 
associated with that in Eq. (5) becomes 
For near side 
Its largest values appear for tz  = 6.5 (this value is determined by the difference t - L). 
Because of the coupling in Eq. (9), this corresponds to a peak in deuteron partial wave near 
el = 2.5, a central collision. The picture obtained from this amplitude is that of a head-on 
collision between the deuteron and the nucleus that gives rise to back-to-back neutron- 
proton separation with the proton emerging on the near side. The sense of rotation in the 
neutron orbit must be the same as for the far-side amplitude, or else the two pieces will 
not add coherently as needed to generate an interference pattern. 
One reason this near-side amplitude is missing from our standard reaction model 
involves the large relative n-p momentum present at the scission point. For the far-side 
contribution, k,, -- 0.6 fm-l, a value still within range of the bound-state deuteron 
internal momentum. That this value is large indicates that even for t = 4 transfer, the 
neutron does not have enough angular momentum in the bound state to make this reaction 
well-matched in angular momentum at the nuclear surface. For the near-side amplitude, 
the relative momentum must increase to k,, -- 1.3 fm-l. The smallness of the bound- 
state deuteron wavefunction for this value of k,, insures that it will be negligable in a 
distorted wave model. One possible explanation that is under investigation is the inclusion 
of break-up through a coupled-channel calculation, with the hope that the intermediate 
breakup state would have sufficient strength at largervalues of k,,. In a recently published 
review, Yahiro shows momentum distributions for breakup into relative S- and D-waves (see 
Fig. 16 of Ref. 5). While the D-wave distribution peaks between k,, = 0.3 and 0.4 fm-l, 
it too is essentially zero by k,, = 1.3 fm-l. Additional calculations are being made with 
the hope that higher internalk angular momentum states may prove to be the appropriate 
intermediate step. 
1. V.R. Cupps, J.D. Brown, C.C. Foster, W. P. Jones, D.W. Miller, H. Nann, P. 
Schwandt, E.J. Stephenson, and J.A. Tostevin, Nucl. Phys. A469, 445 (1987). 
2. E.J. Stephensson, V.R. Cupps, J.A. Tostevin, R.C.Johnson, J.D. Brown, C.C. Foster, 
W.P. Jones, D.W. Miller, H. Nann, and P. Schwandt, Nucl. Phys. A469, 467 (1987). 
3. E.J. Stepheson, A.D. Bacher, G.P.A. Berg, V.R. Cupps, D.A. Low, D.W. Miller, C. 
Olmer, A.K. Opper, B.K. Park, R. Sawafta, and S.W. Wissink, contribution to this 
report. 
4. E.J. Stephenson, R.C. Johnson, J.A. Tostevin, V.R. Cupps, J.D. Brown, C.C. Foster, 
J.A. Gering, W.P. Jones, D.A. Low, D.W. Miller, H. Nann, C. Olmer, A.K. Opper, 
P. Schwandt, J.W. Seubert, and S.W. Wissink, Phys. Lett. B 171, 358 (1986). 
5. M. Yahiro, Y. Iseri, H. Kameyama, M. Kamimura, and M. Kawai, Prog. Theo. Phys. 
Suppl. 89, 32 (1986). 


Nature of the Missing Near-side Amplitude in Calculations of Intermediate Energy (d,p) and (p,d) Reactions

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Nature of the Missing Near-side Amplitude in Calculations of Intermediate Energy (d,p) and (p,d) Reactions

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