Transfer reactions and optical potential ambiguities for light heavy-ion systems

Nuclear Physics A505 (1989) 103-122

North-Holland. Amsterdam

TRANSFER REACTIONS AND OPTICAL POTENTIAL AMBIGUITIES FOR LIGHT HEAVY-ION SYSTEMS*

G.R. SATCHLER

Oak Ridge National Laboratoryi, Oak Ridge, TN37831-6373, USA

Received 19 July 1989

Abstract: We investigate the extent to which measurements of transfer reactions between light heavy

ions might provide a signature of a partial transparency in their mutual optical potential. This

transparency allows some contribution from impact parameters smaller than the grazing one. The

‘2C(‘zC, ‘3C)“C ground-state transition is used as an example. Optical potentials are derived from “C + “C elastic scattering at energies from E/A = 20 to 85 MeV and used in DWBA calculations

of the transfer. At the lower energies, potentials which provide similar elastic scattering at forward

angles also predict very similar peak cross sections for transfer, but the angular distribution away

from this main peak depends upon the degree of absorption present. The sensitivity increases as

the energy is raised, so that at E/A = 85 MeV even the peak cross section becomes very dependent

upon the optical potential used.

1. Introduction

Recently, measurements of nucleon transfer induced by 1sO+28Si at E/A = 20 MeV were used ‘) to distinguish between so-called “surface transparent” and

“strongly absorbing” optical potentials which otherwise gave equally good fits to

the available elastic scattering data for this system. In that case, the surface trans-

parent option was found to be unsuitable because it yielded spectroscopic factors

that were too small. Here we examine the possible role of transfer measurements

in resolving an analogous type of ambiguity that often occurs in optical model

analyses of data for light systems such as “C + 12C or I60 + 12C.

The present question is whether strongly or weakly absorbing potentials are correct

for these systems. (“Moderate” or “incomplete” absorption might be more appropri-

ate than “weak”, but we continue to use the latter term for emphasis.) Transfer

reactions involve form factors that are large in the nuclear interior, so they are

sensitive to the degree of damping in the elastic waves at short distances ‘32) and

may provide information on the potentials that generate them.

Such an ambiguity may be resolved by analysis of accurate and sufficiently

extensive elastic data, but this usually entails knowing small cross sections at large

angles, for example in the so-called “nuclear rainbow” region 2Y3). There are examples

known, notably 12C+ 12C at 140 and 159 MeV, where the data are sufficiently

l Dedicated to the memory of Dr. Lionel Goldfarb.

t Research sponsored by the Division of Nuclear Physics, US Department of Energy under contract

DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc.

0375-9474/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

104 G. R. Satchler / Transfer reactions

extensive (out to 90”) that the type of potential is determined uniquely and only

the relatively weakly absorbing one can give satisfactory fits to the data at large

angles 4,5). Further, the weak-W option is clearly desirable 435) in order to provide

a consistent description of 12C+ “C and 160+ “C data at various energies in the

range 10~ E/A< 100 MeV. Nonetheless, supportive evidence from transfer reac-

tions would be valuable and useful. For example, a signature may appear in the

angular distributions for transfer reactions at smaller angles where the cross sections

are more amenable to measurement. We shall see that this is most likely to occur

at the higher energies.

2. Elastic scattering

2.1. INTRODUCTION

Here we review some of the results of optical model analyses of elastic scattering

data, using “C + 12C at various energies as examples to illustrate some ambiguous

situations that might be encountered. First, for orientation we note that strong

absorption in elastic scattering may be typified by an S-matrix whose elements for

the lowest partial waves have moduli IS,1 of order s 10d4, say, while weak absorption

(in our context) corresponds to much larger moduli, b 10m2, say. (We note, however,

that (SLI as large as 0.1 still gives a transmission factor TL = 1 -IS,/‘= 0.99, or 99%

absorption.) Absorption could be reduced because of strong reflection occurring at

an abrupt potential surface which deters the partial wave from entering a strongly

absorbing region, but here we are concerned with potentials that have quite diffuse

surfaces and do not produce strong reelection. Rather, their imaginary part remains

relatively weak even in the interior. For the cases considered here, typically this

means Woods-Saxon potentials with imaginary strengths W - 25 MeV, say, com-

pared to W P 60 MeV, say, for strong absorption, and associated with deep real

parts, with V- 100-300 MeV, say.

Optical model analyses of a particular elastic scattering data set may yield good,

or acceptable, fits with either shallow or deep imaginary potentials. To illustrate

this, data sets 236-R) for 12C+ “C at various energies were reanalyzed using a six-

parameter Woods-Saxon potential and gridding on the value of W, each time

adjusting the other five parameters for optimum agreement with the data. Thus

x’(W), the usual goodness-of-fit criterion, was obtained as a function of W. Of

course, the character of x’(W) will depend upon both the quality and range of the

data, as well as the distribution of uncertainties associated with them. The experi-

mental (statistical) uncertainties usually increase as the scattering angle increases,

thus giving less weight to the important large-angle data. For this reason, the x2

here were evaluated using uniform uncertainties, with some minor exceptions. The

experimental uncertainties “) for the largest and next to largest angles at E/A =

85 MeV are particularly large, so the uncertainties used here were increased by

G.R. Satchler / Transfer reactions 105

factors of 8 and 2, respectively. Similarly the uncertainty at the largest angle for

E/A = 25 MeV was increased x 10. The error associated with the datum in the sharp

minimum at 5.3” was also increased x10 to account crudely for the finite angle

spread of the detectors.

The x2( W) obtained are displayed in fig. 1 for 12C + 12C at various energies; these

results complement others obtained by Brandan 4,5) for both “C + “C and I60 + “C.

2.2. OVERALL BEHAVIOR

A variety of behaviors for x’(W) is seen in fig. 1; the potential parameters at the

various minima are listed in table 1, and the various fits illustrated in fig. 2. (The

S-wave values IS,1 are also listed in table 1, as an indicator of the degree of interior

absorption. Values of Jv, the volume integral per interacting pair, and the r.m.s.

2.01 ,LiyJ!-y 0 20 40 60 80 100 120

W (MeV)

Fig. 1. Variation of the optimum ,yz with W when the other five optical parameters are varied to fit the

elastic scattering of “C+ ‘*C. The labels indicate the bombarding energy E/A in MeV, and refer to the

potentials in table 1.

106 G. R. Satchler / Transfer reacfions

TABLET

Optical potential parameters for “C+ “C

Poten- V W

tial (MeV) (;G) (y;) (MeV) (;E)

fm3)

E/A=20MeV 20AC) 175 0.589 1.071 19.1 1.190 0.585 1371 1.6( -2) u 258.3 4.488 20B”) 175 0.720 0.829 27.7 1.118 0.646 1405 4.7/-3) 2.12 296.6 4.001 2oc 175 0.561 1.153 118.8 0.900 0.645 1414 36-5) 1.82 242.9 4.7 12 201) 209.7 0.512 1.127 20.0 1.169 0.601 1365 IS-Z) 0.82 264.4 4.551

E/A = 24 MeV 24A 150.9 0.630 1.071 16.1 1.240 0.596 1403 3.1(-2) a 250.9 4.559 24B 180.7 0.677 0.920 26.3 1.101 0.716 1436 8.7(-3) 1.06 293.5 4.176 24C 1010 0.362 0.883 120 0.782 0.796 1462 1.6(-5) 2.04 527.1 3.509 24D j&I 0.732 0.834 &!J 0.775 0.801 1460 1.5(-5) 2.17 318.0 4.043

E/A=ZS MeV 25A 218.4 0.641 0.834 44.7 0.924 0.823 1424 l.S(-3) jJJ 289.3 3.843 25B 408.5 0.526 0.831 57.5 0.868 0.838 1432 l.l(-3) 1 .os 363.3 3.605 25c 646.3 0.469 0.802 120 0.710 0.862 1443 5.0(-5) 1.07 446.5 3.409

E/A=30MeV 30A 174.8 0.649 0.854 29.8 1.019 0.675 1232 l.ct-2) 9 242.8 3.919 30B 143.2 0.591 1.009 15 1.192 0.554 1213 5.6(-2) 2.53 197.6 4.290 3oc 335.4 0.554 0.820 &I 0.818 0.733 1246 1.6(-4) 1.13 325.0 3.623

El A = 85 MeV ‘)

85A 129.4 0.681 0.913 47.9 0.918 0.622 997 l.O(-2) u 211.2 4.163

SSB 173.6 0.537 1.009 28.5 0.956 0.750 1038 5.9(-2) 1 SO 202.6 4.19s

85C 49.9 0.934 0.742 150.4 0.262 1.201 1124 3.7(-3) 1.64 167.1 4.798

85D 122.9 0.713 0.867 42.5 0.946 0.620 1010 1.5(-2) (1.72) ‘) 211.3 4.095

SSE 120.3 0.717 0.835 33.2 0.962 0.696 1045 3.5(-2) (1.01)s) 203.1 4.011

The usual Woods-Saxon potential was used, -V(e’v+ I)-‘-iW(e”w+l)-‘, with x, =(r- Ri)/ai,

Ri = r, x 2 x 12’j3. The Coulomb potential between two uniform charge distributions 14) with radii 3.17 fm

was included. Parameter values underlined were kept fixed during the search.

“) Notation X(n) means X x 10”.

‘) Chi-squared relative to minimum value (A potentials), as shown in fig. 1.

‘) Similar to the potentiai of Bohlen et al. 2). d, Similar to the potential of Brandan et al. s).

“) Relativistic kinematics used.

‘) Actual x2 per datum when experimental uncertainties *) used.

“) Actual x2 per datum when experimental uncertainties s) used, except at 3.1’ and 3.3”, where they

were tripled.

radius, of the real potential are included as well, although these quantities only

have significance when the potential has some transparency.) In some cases, distinct

families of potentials are found (as at E/A = 20 and 25 MeV), so that x2( W) breaks

up into discrete segments with a degeneracy at the points where the segments cross.

In other cases, there is a smooth, if sometimes rapid, transition in the parameter

G.R. Sarchler / Transfer reaclions 107

(a) E/A= 20 t&V fbJ WA=WMeV

- 24A

- 0 10 20 30 40 50 60 70 0 IO 20 30 40

%.m.(deg) 0 c.m.@Q) IO'

(eJ ElA=85MeV

10-3 0 5 10 15 20

8,,,,.Wegt

Fig. 2. Fits to the elastic scattering data given by the various potentials of table 1 at the energies E/A

of (a) 20 MeV, (b) 24 MeV, (c) 25 MeV, (d) 30 MeV and (e) 85 MeV.

values from one family to another, as at E/A = 24 and 85 MeV. At all energies,

there is a flat region extending to large W that represents an Igo-type of continuous

ambiguity 9, in which only the surface features of the potential remain unchanged

and there is strong absorption in the interior. [Members of this family, with W =

600 MeV, were originally chosen by Bohlen et al. ‘) at E/A = 25 MeV.] These

potentials, representative examples of which are included in table 1 (potentials C),

108 G.R. Satchier / Transfer reactions

result in the smooth, exponential-like fall in cross section at the larger angles known as “farside dominance” lo).

The data tend to favor the more shallow W fits, which allow additional refractive effects like an incipient rainbow 3*7,10), except at E/A = 25 MeV where, although the data ‘) cover the largest range of angles, they do not show any such preference. (We note that seemingly there is some incompatibility between the E/A = 24 and 25 MeV data sets 6,7); despite their closeness in energy, an optimum potential for one does not give a good fit to the other.)

2.3. INDIVIDWA~ CASES

The x”( W) for the data 2,6) at E/A = 20 and 24 MeV show marked preferences for weak W -20 MeV; indeed, the latter show two such minima with W = 16 and 25 MeV. The curves shown for E/A = 20 MeV were obtained with a fixed real depth V = 175 MeV; when this was done, a secondary minimum (potential 20B) appeared, associated with a different potential family that has somewhat different properties. This was noted and analyzed elsewhere ‘I). When V was allowed to vary also, this secondary minimum was no longer found, but the remainder of the x”(W) curve retained the same features as seen in fig, 1, with potential 20D at the W = 20 MeV minimum. Fig. 2a shows that the secondary potential 20B reproduces the main features of the data, although it predicts minima between 10” and 22” much deeper than those measured; this is the main reason that the associated x2 is twice that for potential 20A. Further, 20B gives the farside/nearside crossover lo) near 1 lo, while the other potentials 20A and 2OC place it at a smaller angle. The more strongly absorbing 2OC gives almost as good a fit as 20A out to 45”, but beyond that it underestimates the cross sections.

The two minima in x’( W) for E/A = 24 MeV provide equivalent fits to these data (fig. 2b). The more weakly absorbing potential 24A allows the farside/nearside crossover to appear at a smaller angle than 24B, analogous to the behavior of 20A and 20B. The strongly absorbing 24C ( W = 120 MeV) also fits quite we11 except for angles beyond about 33”. (24C has a very deep real part: V= 1010 MeV. However, the potential 24D in which we constrained V== 180 MeV gives essentially the same scattering.) There is a striking difference in behavior in x’(W) between E/A =

24 and 25 MeV, with the latter showing only a very shallow minimum for W = 45 MeV and V=218, while close by is another for a different family with W = 57.5 and V = 409. This seems not to be simply due to the higher energy data ‘) extending out to 70” (see fig. 2c) and thus determining the potential better. The optimum potential 25A gives very poor agreement with the E/A = 24 MeV data “) even at the forward angles, with the oscillatory pattern displaced to larger angles by about 5% and the peaks falling off faster with angle than the measured ones. [There are also significant differences between the E/A = 25 MeV data of Bohlen et a!. ‘) and the more limited results of Sahm et al. jl) at the same energy.] Whatever the cause of these differences,


it will be interesting to compare the predictions for transfer using the potentials

from these two data sets.

An attempt was also made to find a single potential which would simultaneously

fit the E/A = 20 and 25 MeV data for Bohlen et al. *.‘). Neither data set could be

fitted in an entirely satisfactory manner in this way, although a compromise could

be obtained with potentials closer to 20B than to 25A.

The analysis of the E/A = 30 MeV data “) led to a x’(W) curve (fig. 1) with a

very shallow minimum near W = 30 MeV (potential 30A), followed by a continuous

Igo ambiguity at larger W. (It is interesting to note that when a folding model ‘) is

used for the real potential, so that its shape is fixed, the x’(W) shows a sharp and

well-defined minimum at W- 19 MeV, similar to that seen in fig. 1 for E/A =

20 MeV. An Igo-ambiguity, with similar x2, develops for W 3 80 MeV.) The weak

and strongly absorbing potentials give equivalent fits (fig. 2d). In this case, we also

illustrate the consequence of fixing the imaginary depth at a weaker value of

W= 15 MeV (potential 30B). Although the corresponding x2 is 2.5 times as large

as at the minimum, fig. 2d shows that the agreement with the data is not bad.

2.4. THE ENERGY E/A=85 MeV

The relativistically correct center-of-mass kinematics ‘*) were used for the energy

E/A = 85 MeV. (This corresponds to using an effective c.m. energy of 496.84 MeV

and an effective mass of 12.270 u in the nonrelativistic Schrodinger equation.) The

optical model search using the E/A = 85 MeV data “) results in a well-defined

minimum (85A) for W = 48 MeV, with a secondary minimum (85B) for W = 28 MeV.

These are followed by a continuous Igo ambiguity for W 2 65 MeV. The latter

corresponds to a different family of parameter values, in particular with real depths

V - 50 to 60 MeV, together with smaller rw and a large a, = 1.2 fm. Actually there

is an extremely shallow minimum with W = 150 MeV (potential 85C in table 1).

The real part of 85C, although much shallower in the interior, is similar to the other

potentials at large radii, r 2 5 fm, say. On the other hand, because of the small

radius, the imaginary part is considerably weaker in the surface (weaker than 85A

for r 2 2 fm) but much deeper at the smallest radii (reaching -110 MeV at r = 0).

As a consequence, IIm UI < IRe UI for r 3 2 fm, but with a strongly absorbing core

at small radii, and this potential 85C could be characterized as “surface trans-

parent” lo).

The properties of the E/A = 85 MeV potentials are reflected in the corresponding

S-matrix elements, as shown in fig. 3 for their magnitudes ISL1.

Both 85B and 85C give poorer agreement with the data than 85A (fig. 2e). The

use of other data uncertainties to evaluate x2 was also explored to some extent. For

example, using the experimental uncertainties “) led to a similar minimum in x’(W)

at W = 42.5 MeV (potential 85D), but with no secondary minimum at smaller W.

Potential 85D gives a better fit to the cross sections on the peak between 7” and 9”

110 G.R. Satchler / Transfer reactions

E/A=85MeV

1.6

Fig. 3. Magnitudes of the partial wave S-matrix elements for ‘*C + “C at E/A = 85 MeV for the potentials

GA, B, C of table 1. (a) elastic scattering; (b) the L’= L elements for the ground-state transfer reaction for angular momentum transfers of I = 1 and 2.

than does 85A, but elsewhere is essentially equivalent. The result of giving less

weight to the deep minimum near 3” was also explored, as a rough way of accounting

for the finite angular resolution of the data. Potential 85E is one example; except

for a somewhat deeper minimum at 3”, it gives closely the same elastic scattering

as 85D.

2.5. SUMMARY

In summary, individual elastic data sets possess idiosyncracies (including the

range of angles covered) that make difficult the inference of a single, global, potential,

G.R. Sadder / Transfer reactions 111

although certain systematic features can be observed. There is consistently a prefer-

ence for potentials with relatively weak imaginary parts, with values of W ranging

from about 20 to about 45 MeV. In some cases, there are two such weakly absorbing

possibilities. Except at E/A = 85 MeV, these can be characterized by the angle at

which the farsidelnearside crossover lo) occurs; the more weakly absorbing of the

pair allows a larger farside amplitude, producing the crossover, and hence the

deepest minimum, at a smaller angle than the other potential ‘I). The data seem to

suggest that the larger angle is the correct one. Further, in each case there are

strongly absorbing solutions which exhibit an Igo-like ‘) continuous ambiguity,

although, except at E/A = 25 MeV, these give poorer agreement with the data. We

now investigate the extent to which nucleon transfer data may be expected to help

further differentiate amongst these potentials.

3. Transfer reactions

3.1. INTRODUCTION

We now compare the predictions for one-nucleon transfer reactions using the

various optical potentials just described. The calculations were made in finite-range

DWBA, including the core-core, or “indirect”, interactions 13), using the program

PTOLEMY 14). However, it should be emphasized that a study of transfer reactions

does introduce some additional uncertainties beyond those associated with the

elastic optical potentials.

3.2. NONLOCALITY

An important uncertainty for our purpose is that the model optical potentials

being used are local, so that each distorted wavefunction should be multiplied by

a (complex) “Perey factor” F(r) which corrects for the nonlocality of the true

potentials that the model ones represent 13). This nonlocality (=momentum depen-

dence) appears as an energy dependence in the equivalent local potential. In

addition, there may be some intrinsic energy dependence in the potential 13), so that

the equivalent local potential has the dependence U(r) = U(r; E(E), E), where

E(E) = h2k2(E)/2p. In the WKB approximation, the Perey factor is given by 13)

F(r; E)=[l-aU(r; E, E)/ae]“‘,

which exhibits the general property that F + 1 at large radii. In practice, this

expression does not uniquely define the F to be used with a model potential U( r; E)

because we cannot separate the empirical energy dependence into its intrinsic and

nonlocal components without some further assumption.

A simple (Perey-Buck) form of nonlocality that is frequently assumed 15) results

in a damping of the wavefunction, F(r) < 1, inside the potential. Microscopic

calculations of the real potential for 160 + I60 have been made 16) using the resonating

112 G. R. Satchler / Transfer reactions

group method (RGM). The RGM introduces explicitly the nonlocality due to antisymmetrization between the nucleons in the two nuclei. It was found that the corresponding Perey factors produced damping at small radii (for ri5 3.5 fm in this case), plus a small enhancement at larger radii. The damping was glO% for E/A% 25 MeV, the energy region of most interest to us, but became considerably stronger as the energy decreased.

Additional nonlocality can arise from couplings to nonelastic channels. A recent study “) indicates that this may lead either to damping or to enhancement. Con- sequently the Perey effect may either reduce or magnify any contribution to the transfer amplitude due to incomplete absorption in the optical potential, although the present indications are that the effects are not sufficiently large to obscure the results of weak absorption at the energies that concern us here.

3.3. EXIT CHANNEL PARAMETERS

Another uncertainty arises because the exit channel in a transfer reaction differs from the entrance one, and usually the corresponding elastic scattering is not susceptible to measurement. The standard procedure, that is also followed here, is to use the same optical potential parameters as in the entrance channel, implying the assumption that radii scale like (A, “3 + A:‘“]. This is not unreasonable since the residual nuclei differ from the initial ones only by the transfer of one nucleon, and we expect the potential parameters to vary smoothly and slowly as the mass numbers change. For example, data 2, for 12C + 13C at E/A = 20 MeV have been examined in a like manner, and the results found to be very similar to those for “C+ 12C. However, the couplings to other channels will be different for the residual pair of nuclei so that some modification of the potential might be expected; for example, in our case the entrance channel is for two identical even nuclei, while the residual nuclei are both odd and distinguishable.

3.4. VALIDITY OF DWBA

In addition, we rely upon the validity of the DWBA, thereby neglecting any possibility of multi-step contributions to the transfer. This also assumes that the one-channel optical potential provides a good description of the elastic wave in the nuclear interior, although it is known that some changes can occur when nonelastic couplings are taken into account explicitly r7). Of course, these objections may be met by making elaborate coupled-channels calculations, including both inelastic and transfer channels 13), but this was not done here.

It has been suggested “) that for our present purposes the validity of the DWBA prescription may be tested by making transfer measurements in a case where an essentially unique optical potential can be determined from the elastic scattering. A good example would be a “C + r2C at 159 MeV [refs. 4,5,1’)].

6. R. Satchkr / Transfer reactions

3.5. ROLE OF TRANSPARENCY IN TRANSFER REACTIONS

113

Transparency in the interior means, in the present context, that the magnitudes IS,\ of the elastic S-matrix elements are not vanishingly small for low L-values. We can use the Sopkovich prescription *‘) to understand how the transfer amplitudes are affected. The effect of the nuclear optical potentials on the DWBA transition amplitude between an initial partial wave, Li, and a final one, L,, is approximated by modulating the amplitude in which only Coulomb distortion is taken into account by a nuclear distortion factor

E $$J’2 1 (11 where the SL are the elastic S-matrix elements in the respective ehannels. Thus the degree of absorption for the low-L waves is directly reflected ‘) in the magnitude of the transition amplitudes for these ~5, while S,+ 1 for large L so that these contributions are determined by the Coulomb field alone. Between these two extremes are the peripheral partial waves which are sensitive to the degree of surface absorption “).

4. Applications to transfer reactions

We now use the results of sect. 2 to examine the possible consequences of the ambiguities discussed there. [Sometimes transfer measurements have been made at an energy where the elastic data were not available 19>. It has then been necessary to extrapolate optical parameters from other energies. Usually this results in potentials that do not give closely the same elastic scattering. This situation is not considered here because it illustrates the possible results of a lack of knowledge rather that a true optical model ambiguity.]

We chose to study the “C( “C, 13C)‘1C, pII + p3/2, ground-state transition, which involves angular momentum transfers 13) of 1= 1 and 2. The neutron binding potentials were taken to be of Woods-Saxon form with r, = 1.25 fm, a = 0.65 fm and a spin-orbit term of strength Vs.,. = 7 MeV. The differential cross sections are propor- tional to the product of target and projectile spectroscopic factors r3) C:S, , C&T,,

which we denote simply by S. Empirical values 19) range from S = 1.7 to 3.

4.1. TRANSFER AT E/A=85 MeV

The most dramatic effects are seen at E/A = 8.5 MeV. Unfortunately there are no transfer data available at this energy, but perhaps the results presented here will encourage the making of such measurements.

The potentials deduced from the elastic data are typified by 85A, B and C {table l), representing strong absorption, weak absorption and surface transparency, respectively. The transfer angular distributions obtained when using them are shown in fig. 4. The curves are normalized to 10 mb/sr at their peaks, corresponding to


0 5 10 15

6 c.m. fde@

Fig. 4. Transfer cross sections at E/A = 85 MeV predicted using potentials 85A, B, C. The theoretical

curves are normalized to 10 mb/sr at the main peak; this required spectroscopic factor products of S = 2.0

(85A), 1.6 (85B) and 0.93 (8X).

spectroscopic factors of S = 2.0 (MA), 1.6 (85B) and 0.93 (85C). Thus the largest

cross section is obtained with the surface transparent potential, as expected ‘).

These predicted transfer angular distributions differ strongly in shape as well as

magnitude, much more so than the corresponding elastic ones (fig. 2). Decomposition

into the (incoherent) natural (1 = 2) and unnatural (1 = 1) parity contributions shows

that they have comparable magnitudes, very different angular distributions, and that

both are strongly affected by the optical potential used. The decomposition is

illustrated in fig. 5 for the optimum potential 85A.

The partial wave distributions of the transfer S-matrix elements S:., for L= L’

are included in fig. 3, where they may be compared with the co~esponding elastic

ones*. The surface transparency of potential 85C shows up as a strong enhancement

of the transfer for peripheral L-values, and hence a larger peak cross section, while

the greater “volume” transparency of potential 85B leads to an enhancement of

transfer for all partial waves below the grazing one, L%60. The latter effect is

responsible for the large increase in cross section at the larger angles when this

potential is used.

In this particular case, the elastic data seem to be complete enough, and sufficiently

precise, that the potential 85A (or the equivalent 8_5D and 85E) is determined as

* Considerable cancellations are encountered in the numerical evaluation of the radial integrals for

the lower L with potentials 85B and C, resulting in great sensitivity to the integration parameters used.

The curves shown in fig. 3 are averaged over the fluctuations obtained in the actual calculations. These

fluctuations have negligible effect on the calculated cross sections.


10

-$ B

5 1.0

$ D 7J

0.1

I

1% (‘*cJ3c)“c EIA = a5 MeV

- 1=1+2 -- I=2

0 2 4 6 a 10 12

e Cm. (de@

Fig. 5. Decomposition of the transfer cross sections at E/A = 85 MeV into the contributions from angular

momentum transfers of I = 1 and 2. Potential 85A was used.

the “correct” one. Nonetheless, it would be valuable to have corroboration of this

from observing the transfer reactions.

4.2. TRANSFER AT EfA = 25 MeV

This energy is of interest because (i) there are some data 19), and (ii) the analyses

of the elastic data at E/A = 24 and 25 MeV give quite contrasting results, as discussed

earlier (see fig. 1). First we study the use of the three potentials, 24A, B and C,

obtained from the lower energy elastic scattering. (Potential 24C and D give almost

the same results.) As fig. 6 indicates, the predicted peak cross sections differ by less

than l%, while the differential cross sections fall off faster with angle beyond the

main peak as W increases. (However, once the strong absorption region, W+

80 MeV say, is reached, there is very little change with change in W.)

The peak cross section has been measured “) in this case and implies a spectro-

scopic factor product S = 2.7. The data at larger angles are not sufficiently dense to

select one potential unambiguously, although there is some preference for 24B and

the most weakly absorbing one, 24A, seems to be unfavored. The differences between

24A and the other two curves become greater at angles past 20”, reaching almost

an order of magnitude by 30”, so that measurements here could easily distinguish

between them.

116 G.R. patchier / Transfer rem&m

This case illustrates two points: the importance of measuring over the main peak

so that the overall normalization can be determined, and the importance of precise

measurements at larger angles if optical potentials are to be discriminated between.

Fig. 6 also gives the results of using the three potentials obtained from the

E/A = 25 MeV elastic data ‘). They show the same trend of predicting almost the

same peak cross sections for transfer, and a rate of fall-off beyond the peak that

increases as W increases, although here the differences are much less and do not

become significantly bigger at larger angles. This reflects the more strongly absorbing

nature of the potentials obtained at E/A = 25 MeV. The predicted peak cross sections

are also about 30% larger, so that the spectroscopic factor needed to match the

measured value is reduced to S = 2.1. With this normalization, all three theoretical

curves decrease more rapidly with angle than the measured cross sections.

Overall, the available transfer data at this energy seem to favor the less strongly

absorbing potential, 24B, obtained from the E/A = 24 MeV elastic scattering, rather

1% (‘%,%)“C

WA = 25 MeV

.____ 258 __^

/ , 5

, , I. I, I

10 15 20

0 c.m. (de@

Fig. 6. Predicted transfer cross sections at E/A = 25 MeV using potentials 24A, 8, C (above) and 2SA, B, C (below). The data are from ref. 19).

G. R. Satchler / Transfer reactions 111

than any potential deduced from the E/A= 25 MeV elastic data. It would be of

interest to have more complete measurements of this reaction extending to somewhat

larger angles.

4.3. TRANSFER AT EfA=20MeV

Extensive transfer measurements have been made ‘) at this energy, both for the

$- ground state transition and to the $+, 3.85 MeV state in 13C. Potentials 20A, B

and C were used. (Potentials 20D and A give closely similar results.) Characteristics

like those found at El A = 25 MeV were obtained here also; almost the same peak

cross sections, and the rate of fall-off at larger angles increases as W increases.

Unfortunately the measurements do not extend forward to the main peak of the

angular distribution (at about 3.5” for the ground state transition, and 0” for the $’

state). This leads to some uncertainty in normalizing the theoretical curves to the

data. We chose for the ground state transfer to normalize to the three data close to

9”; this required spectroscopic factors of 2.0 (20A), 2.3 (20B) and 2.5 (20C) and

applies to the curves shown in fig. 7. The data show a distinct preference for the

weakly absorbing potential 20A, and appear to rule out the other two.

Calculations were also made for the $’ excited state. This p3,2 + d5,* transition

allows transfers of 1 = 1,2 and 3, and its angular distribution has almost no structure.

The predictions using the three potentials, shown in fig. 8, differ mainly in the slope

of the angular distribution. A normalization of S = 2.3 was chosen to fit the most

10’

0 10 20 30 40 50 60

8 (de@ c.m.

Fig. 7. Predicted cross sections for the ground-state transfer at E/A = 20 MeV using potentials 20A, B,

C. The data are from ref. ‘).

118 G. R. Satchler / Transfer reaclinns

1 o-2

EJA = 20 MeV

0 10 20 30 40 50 60

0 cm. We@

Fig. 8. As fig. 7, but for the transition to the 3.85 MeV t’ state in 13C.

forward data. Potential 20A gives the best agreement below 20” but, in contrast to

the ground state transition, 20B agrees better at larger angles. However, it should

be remembered that 20B gives the poorest fit to the elastic scattering (figs. 1 and 2)

and, indeed, this solution is only found when V = 175 MeV is fixed.

In conclusion, a weakly absorbing potential is needed to explain the transfer data

at this energy. The ground state transition definitely favors potential 20A, but the

excited state one is ambiguous. The strongly absorbing 20C does not agree with

either transition.

4.4. TRANSFER AT E/A=30,35 AND 50 MeV

There are transfer data available 19) for E/A = 35 and 50 MeV, but not at E/A = 30 MeV. Elastic data exist for E/A = 30 and 35 MeV, although the latter “) are

quite limited (3”~ 8% 12”) and could not be used to determine more than a few

parameters.

Three potentials were obtained from the E/A = 35 MeV elastic data, two with

IV = 25 and 80 MeV based upon potentials 30A and 3OC, respectively, and one with

W = 200 MeV based upon Sahm et al. 2t). The three resulting transfer cross sections

agree closely at the main peak and differ in about the same degree at larger angles

as those for potentials 24B and 24C shown in fig. 6 for E/A = 25 MeV. The presently

available 19) transfer data, while in qualitative agreement if S = 3 is chosen, are

unable to discriminate between them.

Calculations were also made for E/A = 30 MeV using potentials 30A, B and C.

The results are shown in fig. 9 for a spectroscopic factor S = 1.7. Included in the

G.R. Satchler / Transfer reactions

‘2’2 (‘2C,‘3C)“C E/A = 30 MeV

s = 1.7

I E/A = 25 MeV 4 EIA = 35 MeV

\ \ ‘..

‘\ 10-l 1 ’ ’ ’ ’ ’ ’ ’ * c

119

0 5 10 15 20 25 30

e c.m. (deg)

Fig. 9. Predicted transfer cross sections at E/A = 30 MeV, using the potentials 30A, B, C. The data 19)

shown are for E/A = 25 (rectangles) and 35 (diamonds) MeV.

figure are the measurements taken I’) at the energies of El A = 25 and 35 MeV which

bracket that used in the calculation. These data certainly indicate that potential

30B, in which W was constrained to 15 MeV, allows too much transparency and

thus predicts cross sections that are too large at the wider angles. The data also

show some preference for the weakly absorbing 30A, but their scatter is too large

for any unambiguous conclusion to be drawn.

There are no measurements of the elastic scattering at E/A = 50 MeV, so Winfield

et al. used potential parameters from E/A = 35 and 85 MeV analyses 8*21). They

found equally good fits to their transfer data, but with spectroscopic factors differing

by a factor of 3.2. The more weakly absorbing potential “) gave the more reasonable

value of S = 1.3, compared to the S L- 4.3 needed with the strongly absorbing one 2’).

We confirmed these results, and also made calculations using the potentials 30A,

B and C. These potentials do not give exactly the same scattering at E/A = 50 MeV,

but close enough for our purposes. They give the same transfer cross section at the

main peak to within a few percent. The predictions are shown in fig. 10 (with a

spectroscopic factor of S = 2.0) and compared with the measured cross sections 19).

Again, the artificially constrained ( W = 15 MeV) potential 30B is unacceptable, but

there is good overall agreement with the theoretical curves for the other two. Although

the prejudiced eye may see some preference for the more weakly absorbing 30A,

the choice is far from clear. Unfortunately the main differences between the two

predictions are largest beyond the range of the presently available data. (Very similar

results (but with S = 2.7) were obtained with the potentials that were fitted to the

limited E/A = 35 MeV elastic scattering.)


102t~,“,““i,“‘I~“‘i”,’

1% (‘*c,‘%)“c

E/A = 50 MeV

- 30A ---- 30B

.--- 3oc ‘, .\

‘\ \.

0 5 10 15

e c.m. (deg)

20 25

Fig. 10. Predicted transfer cross sections at E/A = 50 MeV, using the potentials 30A, B, C. The data are

from ref. 19).

We conclude that more extensive

measurements, at this energy should

strong absorption optical potentials.

4.5. SPECTROSCOPIC FACTORS

transfer measurements, together with elastic

be able to distinguish between the weak and

The emphasis so far has been on the angular distribution of transfer as a possible

signature of weak versus strong absorption in the optical potential. The absolute

magnitudes of the predicted transfer cross sections are also important for determining

the spectroscopic factors. We have seen that, except at the highest energy of

E/A = 85 MeV, potentials which give closely the same elastic scattering at the

forward angles also tend to predict the same peak cross section for transfer.

(Consequently it is important to have measurements across this peak.) However,

ambiguity can arise if more than one set of potentials are available from elastic

measurements at different, but similar, energies such as the E/A = 24 and 25 MeV

examples considered here. In this case, the transfer data 19) taken at E/A = 25 MeV

indicate spectroscopic factors, S, of about 2.7 and 2.1, respectively, a difference of

30% (see fig. 6).

Ambiguity also occurs if elastic scattering has not been measured at an energy

close to that used for the transfer reaction, so that potentials have to be extrapolated

from other energies. The E/A = 50 MeV transfer data provide an example. The

original analysis 19) used published potentials 8,21) obtained from elastic scattering

at E/A = 35 and 85 MeV. Spectroscopic factors of S =4.3 and 1.3, respectively,

were derived. These differ by a factor larger than 3. When we used potentials derived


from the E/A = 30 MeV scattering data, the transfer cross sections were reproduced

quite well (fig. 10) with S = 2.0, by either potential 30A or 30C. Similar results were

obtained with potentials which fitted the more limited E/A = 35 MeV elastic

measurements, but now with S = 2.7, a difference of 35%. Consequently we see that

there can be considerable uncertainties in the value of S for transfer when adequate

elastic measurements are not available at the same energy.

Using the results considered here, we conclude that S = 2.1 f 0.3 for the ground-

state transfer. This is somewhat larger than the shell model prediction *‘) of 1.8, as

well as the average value of S = 3.0 x 0.53 = 1.6 deduced 19) from a compilation 23)

of other (mostly light ion) experimental results for “C. However, it should be

remembered that the value of S obtained also depends upon the neutron bound

state parameters used in our calculations.

5. Conclusions and discussion

We have attempted in the present paper to give some feeling for the sensitivity

of transfer cross sections to the optical potentials for light heavy-ion systems, in

particular to look for additional evidence that these systems exhibit a certain degree

of transparency 4*5Y1’). The available data ‘,19) do provide some evidence in support

of this conclusion, but additional measurements are desirable. For example, more

extensive transfer measurements at E/A = 50 MeV (see fig. lo), together with the

elastic scattering, would be useful. The sensitivity increases with increasing energy,

so that, for example, at E/A = 85 MeV different potentials which give rise to quite

small changes in the elastic scattering (fig. 2e) can give dramatically different transfer

angular distributions (fig. 4) in the same angular region, as well as peak cross

sections that differ in magnitude by a factor of two.

Except at the highest energy (E/A = 85 MeV), we find that a set of optical

potentials which give the same elastic scattering at forward angles also predict the

same peak transfer cross section. Differences in the degree of absorption associated

with the potentials are reflected in changes in the transfer angular distributions at

angles beyond the main peak. In particular, the cross section falls off more rapidly

in this region as the strength of absorption increases. Consequently, it is important

to make transfer measurements across the main peak in order to determine the

magnitude of the spectroscopic factors, and to extend the measurements to larger

angles in order to learn about the absorptive nature of the optical potentials. Figs.

6-10 illustrate this, and also indicate that the sensitivity increases as the bombarding

energy is increased.

The results just mentioned can be understood in qualitative terms. The peak

transfer cross section arises mainly from the same peripheral collisions that determine

the diffraction-like behavior (farside/nearside interference ‘“)) of the elastic scatter-

ing at forward angles. These trajectories are not very sensitive to the interior

properties of the optical potential. Trajectories that penetrate to smaller radii, and

thus are sensitive to the degree of absorption there, contribute most to the elastic

scattering and transfer at larger angles.


The degree of separation of these angular regions can be characterized by compar-

ing the angle 6, where the nearside and farside amplitudes cross ‘“) and near which

the deepest minimum in the interference pattern occurs, with the semiclassical

rainbow angle f& associated with the real part of the potential I*). At E/A = 30 MeV,

for example, 63x is about 7”, while eR is about 50”. The rainbow angle moves in as

the energy is increased, until at E/A = 85 MeV it is close to 10” and the “refractive”

and “diffractive” regions overlap. Thus, at this energy the interior part of the potential

makes itself felt even on the main peak of the transfer angular distribution and the

effects become more dramatic (fig. 4).

Thus the sensitivity of transfer to the optical potential, particularly its degree of

absorption, becomes greatest at the higher energies. Unfortunately, measurements

at these energies alone do not tell us about the conditions at lower energies. However,

the results presented here do suggest that useful information can be gained from

careful transfer measurements even at the lower energies.

Finally, we might note the possibility of similar effects being seen in other reactions

between these light nuclei, such as charge exchange 24).

I am indebted to M.E. Brandan and J.S. Winfield for numerous helpful discussions.

I also thank the authors of the various experimental papers for providing listings

of their measured cross sections.

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Transfer reactions and optical potential ambiguities for light heavy-ion systems